Entertainment Center: TVs, Stereos, VCRs and DVDs - Tapping into existing cable?

I put a spare TV in the basement and want to run cable- There is a cable running along the ceiling along the joists that feed the upstairs TV's- There are no "ends" here- just the cable running through the basement. Can I cut that cable and attach a "T" connection to it?
Also- This older TV just has one single COAX connection to it- Can I hook a newer DVD player up to it?
Thanks

Yes, you can cut your coax and add a 'T'. Be careful not to do anything but cut it and strip the ends, or you will not have enough length. Typically, you don't want more than 4 seperate cable runs (4 tvs) inside your house. The signal to the TV you are cutting will be weaker (by 3dB), if you have a fairly strong signal, you won't notice it.

You can use a DVD player by purchasing an "RF modulator" from pretty much anywhere that sells electronics.

edited to add: you can only split the cable if you are using normal analog cable, not digital, or you will need another digital cable box.

Check with your cable provider. A number of years ago I finished my basement and Time Warner came out and wired the drop for not much more than the parts alone would have cost me. That was a real bargain.