I use the grease that comes with Geissele triggers on all my trigger installs. I place it on the sear surfaces as burt indicated but also on the sides of the receiver where the springs contact (before installing the trigger).
Nothing special, just Ballistol or WeaponShield when I clean it. I try not to get it too greasy or oily because it seems to attract and hold more grime that way.
I love absolutely every thing about it...especially the smell. My wife hates it. But I like having a cleaner/lubricant that works on pretty much any possible surface it comes into contact with.
Ballistol on old mil-surp leather slings works wonders ... I’ve even used it on a couple older belts. I am not sure if that stuff damages anything except maybe a little oil stain on a shirt or such. Also it can be used to clean a cut or scrap at the range if you don’t have anything else.
ETA ... oops I forgot I’m using Slip2000 EWL oil and EWG grease on my firearms lately ... I got a bunch of free samples. The oil is about the same as any CLP type stuff but their EWG grease is excellent on contact surfaces like 1911 rails and such.
No.
On my Geiselle I used the stuff that came with it on install. After that they all get whatever falls off the BCG. If I clean the FCG I'll spray something in there to prevent rust.