Directv; can I buy a used / new satellite dish online and use the receiver at my house?

You get all of that stuff for free and set up properly when you sign up for Direct TV. Trying to save a buck by doing it yourself just doesn't make financial sense.

Yes, if you buy the right dish and wiring setup it will work. Which setup you need depends on what receiver they are giving you.

That said, this is against DirecTVs terms of service and if they find out they can ban you from ever getting DirecTV service, and they can cancel and ban your relative from getting service ever again as well. Your relative will also have to pay any early termination fees that may apply if his service gets cancelled (for example if they have 10 months left under contract with DirecTV they will owe $200 in early termination fees, or $20/month for however many months they have left).

The H25 requires a SWM setup. So the easiest thing to do would be get a Slimline dish with SWM LNB and a power inserter. This would be everything you would need to get it up and running.

http://www.solidsignal.com/pview.asp?mc=02&p=AU9-SL3-SWM-B&d=DIRECTV-AU9SL3SWMBPWS-Three-LNB-KaKu-Slim-Line-Dish-Antenna-w-Long-and-Short-Brace-and-SL3SWM-LNB-Combo-(AU9SL3SWMBPWS)&c=DIRECTV Dishes&sku=

You might be able to find it cheaper on your local Craigslist or eBay, but you need to make sure to get the Slimline Dish with a SWM LNB, and a power inserter. Since you would only be using one HD receiver you wouldn't need a splitter.

It will work fine without being hooked up to their whole home system.

Do not hook it up to your home internet service though, as it will call into DirecTV and report its location, which will tell DirecTV that it is being used at a different address and they will investigate/shut off service.

All this said, I really think it would be best just to sign up for service. If the dish is properly installed wind will not be a problem. I live in a very windy location, and it is only a problem for me on the windiest days (40MPH+), and I believe that is only because my dish wasn't installed correctly. I just haven't had time to fix it. My neighbor with DirecTV, and my HughesNet satellite internet (which also uses a similar dish) don't have any issues with wind, both of those dishes were installed correctly.

Also, depending on your state laws you may have a certain window in which you can cancel service without penalty if you don't like it, or have issues.

Both of those dishes will work fine, however I can't tell from the pictures whether or not the LNB (the feed horn thing on the end of the rectangular arm out in front of the dish) is a SWM LNB or not. Ask the seller how many coax ports there are on the LNB (they will need to remove it from the rectangular tube to see). If there is only one coax port on the LNB then it is a SWM lnb and it will work. If there are 4 coax ports on the LNB then it is an old style one and it won't work (at least not without getting more complicated and spending more money).

Or if there is a part number on the LNB you might be able to tell from that (I'm not sure, haven't looked into it that deep before).

This is a SWM LNB:
http://www.solidsignal.com/pview.asp?mc=02&p=SL3-SWM&d=DIRECTV-SL3SWM-SlimLine-Single-Wire-KaKu-Triple-LNB-With-BuiltIn-Multiswitch-(SL3SWM)&c=DIRECTV Dishes&sku=8546300073

This is a old style LNB:
http://www.solidsignal.com/pview.asp?mc=02&p=SL3&d=DIRECTV-SL3-SlimLine-Four-Out-KaKu-Slim-Line-LNB-(SL3)&c=DIRECTV Dishes&sku=

The sellers might not realize if they have a power inserter or not. It would have been plugged into the coax, and into a wall outlet somewhere inside there house. Some people don't realize that it is actually supplying power to the satellite dish (actually it's powering the multiswitch built into the LNB but that's info thats technical stuff that we don't really need to get into).

I guess since the Craiglist dishes are so cheap though you might consider just buying them and buying a SWM LNB seperate if you need to. You can save a decent amoutn of money if you're willing to buy from eBay. Like this has the LNB, power inserter, and a splitter (which you really won't need).
http://www.ebay.com/itm/NEW-DIRECTV-SWM-3-LNB-4-WAY-8-WAY-SPLITTER-21V-POWER-SUPPLY-FREE-SHIPPING-/120985239374?pt=US_Satellite_LNB_Downconverters&hash=item1c2b483f4e

Then you just need to figure out how you are going to mount it.

The SL3 looks at the satellites at 99°, 101°, and 103°, the SL5 looks at those three locations as well as 110° and 119°. 110° is no longer being used by DirecTV in the US, 119° is but it is mostly foreign language programming so if you aren't getting a foreign language package you don't really need it. There are a few music channels on it as well if those are important to you. Be aware though that the 119° satellite can sometimes be harder to get line of sight to, so in most cases DirecTV doesn't use it unless they have to.
Yes Pwer Supply and Power Inserter should be the same thing. Since you are only installing one receiver you won't use the splitter at all.
No you shouldn't really need anything else other than the tools to mount the dish and cut/add fittings to the coax cable. You can use the signal meter in the satellite receiver itself to aim the dish. There are some good videos on how to do it on Solidsignal and YouTube I believe.
I'm not sure mounting to the side of the house would be a good idea. I dont' believe DirecTV allows that anymore, so while it might work fine for you right now, if you end up getting DirecTV service yourself they probably won't use that dish and will install their own on your roof or on a pole in your yard.
You will run a coax cable from the dish to the power inserter, and then another coax cable from the power inserter to the HD receiver. It probably wouldn't be a bad idea to install a ground block in the line before it enters your house too.
Again, I will say that this is against DirecTVs erms of service and I don't
recommend doing it.
You're welcome.

There is a lot more knowledge and information on DirecTV service and installation over at AVSForums' sister site DBSTalk.com. You should think about going over there and reading/posting. Word of caution though, they are much more strict on talking about doing things against DirecTV's terms of service, so I don't know if I would go into too much detail on what you are trying to do. Here's a link to the installation subforum.
http://www.dbstalk.com/forumdisplay.php?f=113

Yes the SL5 will work. For the other questions I suggest checking out DBSTalk.

You know that what you are doing is fraud, right?