Q:

Is confirmation on checkout still relevant?

Is confirmation of products and shipping details still relevant, or it simply adds information load to buyers.
Especially when we think of mobile experiences.  
I know that we use them to prevent errors, but can't we use something like this (just show the cart, then ask details, but always showing the price):

download bmml source – Wireframes created with Balsamiq Mockups
If we have added a confirmation we would have 4 steps instead of 3, and a screen with many information.
So my core questions are:

Do users really double check their information before purchasing?
Is there any case study or research on that topic?   
What is your approach on this? 

A:

I think you need to see Uber app from a UX engineer's point of view. Make purchasing simple, clear, intuitive and elegant. I will be happy to give my users an option to cancel the order if they feel that they had purchased something wrong instead of putting an extra page for confirmation.
Think it this way when you purchase an item you have an intent to do so. What you can do is put a button to go back to the cart in case user needs to make changes to the purchased items. 

A:

Okay, I don't have user research to back me up but ihave 20 years of user experience to help give insight.    

I don't think you need the price on the fill details screen - its
not needed.
I do think people double check before purchasing ONLY if the price is over a certain amount (like over 20$)or they are committing to flying somewhere on a certain date.  
Booking a flight is a common one for having a check info screen but placing a bid on ebay for only 5$ I dont htink you need a confirm is also a good one to look at.  
My approach would be to assess how fast the user wants to complete the transaction vs how much the money means to them. Some people are fine will spending  without confirming.

A:

If your website is WCAG 2 AA, then you need to be concerned with guideline 3.3.4 Error Prevention (Legal, Financial, Data)
This requires one of the following to be true:

Reversible: Submissions are reversible.
Checked: Data entered by the user is checked for input errors and the user is provided an opportunity to correct them.
Confirmed: A mechanism is available for reviewing, confirming, and correcting information before finalizing the submission.

Two of the techniques they suggest are

The confirmation page (G98)
The ability to reverse the transaction as in @Krishna answer (G164)

