Hi just answered you on chat and in my opinion, no I would not do that. If they want to see your site they can look at your profile if you have it on there, and you can ask for feedback, but to use it as a signature, no I don't think that would be good, just my opinion.

No, as it would be considered spamming your website. You can put your website on your profile so that others know what it is when they view your profile.

Search bar is one of widget options in Appearance=>Widgets=>Search
This is a little later covered in the courses but Google Adsense allows you to add a search bar or box to your website, copy and paste to your website html text or text widget on sidebar. Only thing about Adsense features is that you don't click your own, those Adsense ads owned by you, you shouldn't click. Iguess counts as irrelevant click. But at least you can learn how does search box html script looks like!
The jaaxy plugin also has one with iframes, I forgot.
If you choose to make a box around it, which totally optional, try

Go to Appearance and then Widgets. From the left, select SEARCH. You are going to click and drag to the right side under SIDEBAR. Save.

As Deirdre says, you don't have to buy a domain name however, your own domain is always going to rank better than a sub domain extension. :)

You can keep BOTH EXTENSIONS...FOREVER!
You do not have to purchase a domain.