Is it proper to start a sentence with the word that ended the last sentence?

While I'm not aware of a particular grammatical rule that would prohibit this, as a matter of style, I would prefer not to do what you're suggesting there, especially with something like ASP.NET AJAX. I would suggest something like:

Instead, you’ll use a higher-level model called ASP.NET AJAX. This toolkit gives you a set of server-side components and controls that you can use when designing your web page.

You can replace toolkit with something more appropriate if you prefer.

It is a recognised rhetorical device. That said, when you are not going for rhetoric it may be best if you avoided it.

Yes. You just did.

I don't know of any rule forbidding it and even if there was it likely is simply a best practice.

As with most things, avoid it if it makes the sentences hard to read or sound awkward, even if it is technically correct.