I recently damaged my rear derailleur and am trying to find the replacement online. It is a Shimano Tourney SIS Index. I have found several models in the Shimano website that are pretty close matches, but am wondering if there some way to determine the model or is there a hidden product number in the unit somewhere.
Practically there are only a few things that you need to match a rear derailleur to the one it's replacing.
The first is the mounting point - check the mounting to the frame is the same.
The other is the length of the cage - a short cage is typically used for road bikes and medium or long cage for mountain. The longer the cage, the bigger range of gears it can cope with, but the less precise the shifting is. As a rule try to stick to the same length cage, but if not, go longer rather than shorter (Unless you know what you are doing). try to stay with one for the number of gears as the chain, but it is not critical if its one either side. (i.e. an 8-speed on a 7-speed chain should be fine)
So if they're all pretty much compatible - you are probably wondering why such a large range - it's all down to weight, quality of shifting, expected life and price and just plain old marketing...
My advice - head down to the LBS (local bike shop) with your bike (or just the derailleur), and have a chat with them. They will have something off the shelf that will work, and you can discuss cost/quality etc. The LBS should be able to fit a new one in 1/2 hour if you want them to do it. I have had great success purchasing stuff or bikes online, however have also had the odd incompatibility due to not being aware of some seemingly minor detail.
If the end of the old cable is frayed/cut and not soldered, buy a new one. it's just not worth the effort mucking around with the original cable, and they are only a few dollars.
There are a couple of different versions of the Shimano Tourney, I found the TX35 and TX55, both in a bolt on and rear hanger version. The way these mount must match your bike...if you can post a picture, someone here can likely identify it.
You shouldn't have much trouble finding these online, all four versions that I can get are available via Amazon for about the same price, under $20, as they would do at a bike shop...or put another way, you aren't going to save much/any money online so why not support your LBS (local bike shop).
Not that most bike shops probably won't carry these in stock (the larger ones might), but just about any shop should be able to get one in for you in less than a week.
The rest of your question looks to be cut off, I suspect you were starting to ask how hard is it to replace the derailleur and tune the drive train. Not super hard, but you will want to take it slow and not be afraid to get help. Watch a couple of the how-to videos on Youtube and read through some of the answers to rear derailleur questions here.
If you still have questions, come on back.