Q:

Has the Trump administration revoked these listed LGBT protections?

This image is doing the rounds on Facebook:

It quotes a tweet by Donald Trump from early 2016, followed by claims that the Trump administration has performed these actions:

Thank you to the LGBT community! I will fight for you while Hillary brings in more people that will threaten your freedoms and beliefs.
Donald J. Trump – @realDonaldTrump

Jan: Removed all content on LGBT civil rights from whitehouse.gov website
  Feb: Rescinded protections for transgender students on their use of restrooms in public schools
  Mar: Revoked protections for LGBT workers against discrimination in hiring employment
  Apr: drops federal lawsuit over North Carolina's statewide prohibition on LGBT equality
  July: Signals the US military will not "accept or allow" transgender people to serve

Did the Trump administration perform all of the claimed actions, on the months cited?

A:

Jan: removed all content on LGBT civil rights from whitehouse.gov website
True

During Barack Obama’s presidency, if you typed whitehouse.gov/lgbt into your browser, you reached a page highlighting the administration’s victories and policy changes regarding LGBT rights. It outlined historic court victories and even featured campaigns like the It Gets Better Project to help LGBT youth.
Today, however -- just hours after President Donald J. Trump took the oath of office as the United States’ 45th president -- if you type in whitehouse.gov/lgbt, you are redirected to a new “transitionsplash” page.

CBS News

Feb: Rescinded protections for transgender students on their use of restrooms in public schools
True

The Trump administration on Wednesday revoked federal guidelines specifying that transgender students have the right to use public school restrooms that match their gender identity, taking a stand on a contentious issue that has become the central battle over LGBT rights.

Washington Post, see also Reuters, NPR, NY Times.
With regard to the effect of this guidance on court cases, see as an example, this order, which cites both the Obama administration guidance and the Trump administration withdrawal of that guidance as cause for, first, the 4th district to reverse a lower court opinion (the original Obama administration guidance) and then caused the Supreme Court to vacate the 4th district's decision (in response to the Trump administration withdrawal of that guidance).
In response to Obama administration guidance

In a decision dated April 19, 2016, we reversed the district court’s dismissal of
Grimm’s Title IX claim, relying on a guidance document issued by the U.S. Department
of Education and U.S. Department of Justice.

In response to Trump administration withdrawal

After the Supreme Court calendared the case for argument, the new Administration
issued a guidance document on February 22, 2017, that withdrew the prior Administration’s guidance document regarding the treatment of transgender students, and the Court then vacated our April 2016 decision and remanded the case to us “for further consideration in light of the [new] guidance document issued by the Department of Education and Department of Justice.”

Mar: Revoked protections for LGBT workers against discrimination in hiring employment
True - his action was limited to discrimination limits relating to federal contractors.

Trump signed an order on Monday revoking protections signed into law by President Obama in 2014. Obama signed an executive order banning LGBT discrimination among federal contractors; he concurrently signed an order requiring contracted businesses prove they're complying with federal laws and executive orders. President Trump rescinded the latter order, making it much more difficult to know whether a business has committed to ending LGBT bias in hiring, firing, and promotions.

The Advocate, see also Rolling Stone, Boston Globe, Salon

Apr: drops federal lawsuit over North Carolina's statewide prohibition on LGBT equality
True- the reason given for dropping the federal suit is that North Carolina repealed one bill and replaced it with another. The new bill was watered down but still attracted intense criticism by LGBT groups who believe the federal suit should have continued against the replacement bill.

Officials said that they were abandoning the lawsuit because North Carolina lawmakers last month enacted a law repealing the bathroom bill and replacing it with another measure. The new law, however, has prompted intense criticism from the LGBT groups who long opposed the first bill and are vowing to keep fighting the new measure in court despite the Justice Department’s decision to bow out.

Washington Post, see also CNN

July: Signals the US military will not "accept or allow" transgender people to serve
True

Donald Trump said on Wednesday he would not allow transgender individuals to serve in the US military in any capacity, reversing a policy put in place by Barack Obama a year ago.
The US president tweeted: “After consultation with my generals and military experts, please be advised that the United States government will not accept or allow … transgender individuals to serve in any capacity in the US military.”

The Guardian, see also NY Times

A:

Jan - mostly true
Feb - true
Mar - somewhat true
Apr - mostly false
Jul - true

Jan, removed all content on LGBT civil rights from whitehouse.gov website: mostly true
This is rather misleading as every new administration removes and archives the content of whitehouse.gov. You can see the old site at obamawhitehouse.archives.gov.
Redirects have been added for LGBT content, e.g. whitehouse.gov/the-press-office/2016/06/29/fact-sheet-promoting-and-protecting-human-rights-lgbt-persons. It is arguable whether this is still counted as being "on" whitehouse.gov.
The Trump administration has not added any LGBT-related content on the new whitehouse.gov. In fact, the entire website is significantly lighter than it used to be. Compare current and previous issue pages. Perhaps this is due to time in office, preference, or other reasons.
Feb, rescinded protections for transgender students on their use of restrooms in public schools: true
Obama's Education department issued a letter:

When a school [that receives Federal funds] provides sex-segregated activities and facilities, transgender students must be allowed to
participate in such activities and access such facilities consistent with their gender identity.

This court decision over  transgender bathroom use cites that document in its decision.

we reversed the district court’s dismissal of Grimm’s Title IX claim, relying on a guidance document issued by the U.S. Department of Education and U.S. Department of Justice

Trump's Education department revoked the letter, and therefore any legal basis that relied on it.
EDIT: Credit to @DeNovosupportsGoFundMonica for pointing the error of earlier legal opinions I relied on, which stated that Trump (or any other president) could issue opinions but could not actually affect legalities.
For context, I leave the erroneous statements from Lambda Legal and the ACLU below:
This was only ever "guidance."

Trump's actions do not change the law itself -- transgender students remain protected by Title IX of the Education Amendments of 1972 -- but abandoning the guidance intentionally creates confusion about what federal law requires.
Rachel Tiven, CEO of Lambda Legal

and

While it's disappointing to see the Trump administration revoke the guidance, the administration cannot change what Title IX means.
James Esseks, ACLU

Obama's letter itself did not add any legal protections for transgender bathroom use. Rather, it was a statement of what the executive branch desired to enforce.
The courts would decide any such cases independent of the Obama's or Trump's opinion on Title IX.
Mar, revoked protections for LGBT workers against discrimination in hiring employment: somewhat true
Two caveats: (1) it only ever applied to federal contracts, and (2) it was a purely administrative change, not a legal one.
Trump revoked the Executive Order 13673.
It required evidence that suppliers for federal contracts of $500k+ were in compliance with Fair Labor Standards Act, Occupational Safety and Health Act, Migrant and Seasonal Agricultural Worker Act, National Labor Relations, Davis-Bacon Act, Service Contract Act, Equal Employment Opportunity Executive Order, Rehabilitation Act, Vietnam Era Veterans' Readjustment Assistance Act, Family and Medical Leave Act, Civil Rights Act, Americans with Disabilities Act, Age Discrimination in Employment Act, Establishing a Minimum Wage for Contractors Executive order, and "equivalent State laws".
These are still laws, but compliance will no longer have to be demonstrated for every contract.
Importantly, the concurrently issued Executive Order 13672 that actually covered LGBT discrimination remains in force.
And none of this matters for employers who are not federal contractors.
Apr, drops federal lawsuit over North Carolina's statewide prohibition on LGBT equality: mostly false
"Statewide prohibition on LGBT equality" is overly broad on two counts:

The Public Facilities Privacy & Security Act pertains only to transgender persons, not any other LGBT person (lesbians, gays, bisexuals).

The law is scoped strictly to bathroom use in government facilities. It does not affect housing, employment, taxes, etc. All existing protections remain in place for these.

And even more glaring problem with the claim is that North Carolina repealed the law. While technically the Justice Department did perform the legal formality of withdrawing their suit, they did this only because the defendant acquiesced. By any reasonable definition, the Justice Department "won" their case.
When North Carolina repealed the law, they did replace it with another prohibiting local governments from legislating in this area. Though the practical consequences of the new law likewise has the ire of transgender groups, there is really no ground for the Justice Department to claim constitutional violations.
July, signals the US military will not "accept or allow" transgender people to serve: true
Then Donald Trump announced via Twitter

the United States Government will not accept or allow...Transgender individuals to serve in any capacity in the U.S. Military.

For context, at the time of this answer, openly transgendered recruits have never been permitted in the U.S. military. Obama announced the restriction would be lifted, but not until a full year after he left office, in 2018.
EDIT: Note that the claim did not say that President Trump ordered or carried out the ban's continuation; the claim conservately stated that he "signaled" it, and he certainly did that.

