Why wouldn't they?
Never heard of not saying it.
Yes, which surprised me since the place I Davened Mincha yesterday didn't say it at Mincha even. Just assumed most places wouldn't say it today...
It's to show that Moshe rabeninu is no worse than a Rebbe that we skip tachanun on yahrzeit for. Seriously I once heard this from a litvak.
Bifeirush:http://en.wikipedia.org/wiki/Tachanun#Days_on_which_Tachanun_is_omitted
At least in Chabad we say tachanun on Rebbeim's yahrzeits.
Davened for amud at r' rottenberg in monsey. No Tachanun