Yes! Pilgrims are like any other card. The restriction that prevented them from leaving your hand was just a special mission rule for Mission 7. You can play them or discard them to damage as normal for mission 8 onwards.
"Any time" is literally whenever you want.. even in the middle of resolving a class ability - so yes, you can draw a whole bunch of cards with the chanter, throw one on the relic and then discard down to hand size.