pre-reqs

The answers to your questions will vary from school to school. Some require you do the TEAS, some require the HESI. Some want your scores before you apply, and some (I think) will want them at a later time. Some schools have a higher average score than others. For your question regarding the first degree you get, nursing schools will have either an ASN or BSN, and from there you can go on to getting an MSN, PhD, or DNP. But if you're referring to the first degree you get prior to even starting nursing school...well, I don't know of any that need a degree to get in; it's just the pre-requisite courses (unless the school offers an accelerated program, in which case you'll need a bachelors degree in some other field [doesn't usually have to be science-related]).

I looked up TWU and found this. If you scroll down a little, it mentions that there are information sessions at both the Dallas and Houston campuses. It appears that the Dallas one has just passed (though they may have another on in the future), but the Houston one has not yet come about, and that you can sign up to attend using the link on the site page. I'd ask your questions there or look around on the schools site to see if there's any additional information.

Hope that helps.

When to take the test depends on some factors like the deadline of turning in your application to the nursing program and also the amount of time spent studying for the exam. when you're ready to take the exam go for it.

if you're applying to university schools then they should have BSN programs. that's the degree you want. I think depending on your area it may or not be very competitive to get into.