Q:

What is そんでもって?

According to コトバンク, it is the exact same as two different definitions of それで。
Then there is this ベストアンサー which gives many usage examples. Again, it seems to be a very casual filler word meaning それで。On that same page, another person says it is from それで以て and that 以て：手段・方法・材料などを表す so that it means …で。…でもって。…によって。
(1) But if the entire expression just means それで, what is the role of the 以って（以て）? Is it actually adding meaning? I mean, should it actually be thought of as それで＋以って？
(2) Would it be considered slang or standard?
(3) How often is it used and does usage frequency change with age bracket or gender?

A:

「そ（れ/ん）でもって」 means practically the same thing as 「そ（れ/ん）で」.  The only difference worth mentioning IMHO is that the former sounds more emphatic than the latter.
The former is the type of expression that tends to be used (repeatedly) by the same speakers.  Other speakers, myself included, would rarely if ever use it.
「以{もっ}て」 is a unique word in the sense that it has multiple, very different meanings and usages.  Placing emphasis is just one of them.  
Time: 「10月31日を以てSEを辞{や}めます。」 = "I shall leave SE as of October 31." 
Reason: 「交通違反{こうつういはん}を以てドミノピザより解雇{かいこ}となった。」 = "(Someone) has been fired by Domino Pizza for traffic violations."
Method: 「毒{どく}を以て毒を制{せい}す。」 = "Fight fire with fire."  Literally, "Control poison with poison."
Emphasis: 「レレクトゥールの回答{かいとう}は全{まった}く以て意味不明{いみふめい}だ。」 = "@l'électeur's answer is just completely nonsensical."  In this sentence, 「以て」 is used purely for emphasis; therefore, the sentence's meaning will not change if 「以て」 is dropped.  
There are a couple of other advanced usages of 「以て」, but I will not get into that.

(1) But if the entire expression just means それで, what is the role of the 以って（以て）? Is it actually adding meaning? I mean, should it actually be thought of as それで＋以って？

The role is to emphasize 「それで」.  As I stated above, dropping the 「以て」 will not change the meaning of 「それで」 one bit.  For people who do use 「そ（れ/ん）でもって」 regularly, I am pretty sure it is just "one word".
One thing that I kind of hesitated to mention earlier is the "good" rhythm the phrase 「そんでもって」 creates for the native speakers. It sounds lively with both a ん and a small っ in such a short expression.  Sound is most important in Japanese because after all, Japanese has been a spoken language for most of its history.  Sound was all we had.  Need I mention our countless onomatopoeias? 

(2) Would it be considered slang or standard?

It is colloquial, informal, conversational, etc.  I would not call it slang as it is not something understood exclusively by special groups of people.
Use it, if you want to, only with your friends and family in informal situations. 

(3) How often is it used and does usage frequency change with age bracket or gender?

As I mentioned earlier, it is not used by "everyone" -- at least not on a regular basis.  It is often the same people who "habitually" use 「そんでもって」.
Age and gender have little to do with the usage frequency, but you will rarely, if ever, hear a toddler use it. 

