How or why did those Alt-Right psychos choose a frog as their mascot? Frogs are cute; shouldn't be maligned like that!

They didn't really.   It mostly has to do with a misunderstanding of  "chan" meme culture.

Pepe has been a staple 4chan and other chans for a long time and part of "chan" culture is crass shock image humor.    Putting offensive and shocking images up purely for there offensive, absurd, and crass nature.  It's a one-up kind of thing.   Anyways that means a lot of nazi and other horrific imagery again based on the trying to one-up on  the shock value, especially to non-chaners.  

Why Pepe got associated with the Alt-Right has to do with people taking notice of some of these shocking and offensive for the sake of being offensive jokes as being serious and taking it as true racism.   After that the image became embraced by the Alt-right as an image due to the misunderstanding of what it really was at first.  I'm not saying there aren't racist POS in the chan culture, but that's in every loose social circle.  Yet that is how this came to be. 
It started off as a joke and got pushed into the reality.   People will argue with this assumption, but at the same time I doubt they have been truly exposed to, or understand the older internet and chans it came from.