Is there sufficient capacity on Crossrail to run all the trains from the north and west of the UK through to Paris??
I don't think capacity is the problem, it is initially a question of train type. Look at the issues with the border agency re: Amsterdam & Frankfurt to London trains. To be viable anyway, these would join / split in Brussels, where I assume they would also enable BRU pax to (dis) embark.
The market for flights to Paris from BHX is essentially a monopoly tie-up with 6 x approx. 100 seats per day. Also known as half a train load.
So to have any change of working, trains would need to split / join at Brum's M42 station, except that only the hybrid trains can start at Manchester, and will these be capable of going through the tunnel? Come 2032, when phase 2 opens, trains might be able to run through, but given the pressures by then to run clockface timetables, how would they squeeze the slots in? Much easier to run more services on HS1 starting at St P's, which could move to services running at set frequencies - Paris currently 20/day, Brussels 10, but they don't always run at same time past each hour.
So the simplest question would be how to enable an easy transfer between Euston and St P's - again, haven't seen that one in the plans.
Have they built 'sideings' on CrossRail, to allow express trains to pass through, on the otherwise slow (multistopping) CrossRail traffic?
You would never pull a train into a siding to allow a faster one to pass through. That is what stations are for, but no, that just wouldn't work on Crossrail - far too many stops, different purpose.
However, the French do engineer their TGV stations to allow a fast train to come through, but you still have the issue of acceleration and deceleration. That is why there is no HS2 station between OOC and Brum M42 - much as though some of us in Coventry would like one! There are a few of us here who are imbys rather than nimbys!