"how much mail space does this give me" - no idea, good question! I can't seem to find an answer either.
(b) if you wanted to keep it simple, you would want to transfer your domain to Zen (and therefore pay them the yearly domain fee instead of your current registrar), and you would need to purchase some sort of hosting plan. I don't think Zen offer pure email-only hosting plans, so you'd be looking at a Web hosting plan which comes with mailboxes (I think £5.99 per month is the cheapest).
Alternatively and depending on how many mailboxes you want, you could keep your domain with whatever registrar you want, and set up a Google Apps or Office365 plan which would give you email, storage, Office365 if you choose the MS option, etc all on your own @bloggs.co.uk domain. Google Apps works out cheaper unless you also want MS Office, in which case O365 would probably be the better choice. There are plenty of comparisons online.
3) Absolutely, regardless of email provider, you should be able to do this on any of them.
My own, purely personal opinion given that I have a comfortable enough level of technical knowledge: I would keep my domain registered wherever, and set up GApps or O365 for my own domain, which takes care of email etc. I don't like using an actual email client and download emails, I prefer webmail and I'll pretty much always leave the emails on the server.