I can only talk about the first one, and only from my own point of view and opinions. Yes, we are meant to be quiet in church.
The prayers after communion are supposed to be said (and heard) with the same reverence we give to the liturgy - we're thanking God for the gifts He's (he's?) given us, unworthy though we are. I dunno about you, but I can't really be thankful and concentrate on the prayers when I can barely hear them because everybody is chattering (please realize that I know I'm one of the chatterers a lot of the time. My monitor right now has a mirror over it and I'm talking to myself as much as everybody reading this.)
When we talk during service, we distract others from concentrating on the service, on the words and meanings and prayers that are being offered - on our behalf! - to God. So yes, it behooves us to shut up. Besides, I often get the feeling if God decided to answer back, we wouldn't hear Him (him?) for all the noise!
Grump grump grump. I'm being a bad tempered baba.
