Non-locked DVRs on Verizon? HTPC doesn’t work with Windows 10 and IIRC Windows 7 was the end of that. You still need to rent a card from the cable company for that. There is no simple end user way of recording stuff in 2015 using removable media like we were able to in 1980 with VCRs. Even if what you are saying is possible it is definitely not mainstream and seems like you’d need to have to reverse engineer stuff to do what simple VCRs did back in the day.
No, it is possible. The problem is that you can’t use HDMI or DVI as a recording source courtesy of HDCP unless you buy semi-illicit gear from China.
HTPC is still possible under 10, they just dumped the media interface because there are better aftermarket alternatives and nobody bought the HTPC edition of Windows anyway.