I am not even sure that a capture card or DVR (if you can still find one) will work alongside cable any more. All that anti-piracy stuff has pretty much gutted this ability.
Most network shows will be available for streaming from the various online presence of said network within a day or so. The sports events I am not so sure about.
It doesn't, but does.
So the cable box will decode the signal and a PC capture card will be able to see that one channel just fine. It just doesn't allow for the typical DVR capabilities like having it automatically record a different channel. You would have to 1) have the cable box actually on 2) set it to the channel you want to record using the remote. Really only useful for specific recordings.
A PC capture card can still be used to grab ClearQAM signals as mentioned, which there should be quite a few on the raw cable without the box. Public access and local stations are often not encrypted since they just take the over the air broadcast signal and re-transmit it via cable. But you won't be able to see the paid cable channels.
It just doesn't make sense anymore. Not to mention that all the cool tools aren't supported. Windows Media editions actually had decent integrations for capture cards, I think that worked up through Windows 7 with some hacking. Didn't try it after that myself, I ditched television service entirely about 8 years ago now. Though I still technically have streaming TV via my phone plan, I don't use it. Just let the other members on the plan use it.
I would almost say Youtube TV might be the cheap option these days? Free cloud based DVR.