I try to reply you.
Yes, Geexbox can capture the tuner signal and record all TV programs. You need just to check if a firmware for your driver exists for linux and derivates… You can’t use windows drivers.
Yes, Geexbox/tvheadend can schedule recordings based on EPG grabbed in internet.
Yes, Geexox can generally use remotes for receive command. You need to check if this is true also for your own remote. If you are luck you can also found a table already done for mapping keys of your remote to XBMC commands. Otherwise you can configure it by hand. If you also use a CEC compatible TV you can also use same remote for TV and Cubox. I have a Samsung Smart TV and it works fine.
Let me know if I replied to all your questions.