mirror site questions - please advise!

I have some concerns about my site at www.abuseneglect.com. If you
look at the source code, you can see that this site points to
http://www.nursing-home-abuse-neglec...nformation.com. We use the latter
URL because it is keyword-intensive and has the potential to do well
with the search engines (we are just getting going on SEO for this
site). The shorter URL, though, is much easier to give as a reference
to our potential clients (i.e. "visit our site at..." - it would be
cumbersome to recite the longer URL and hard for them to remember).

I have a few questions about this. First, is this ok, to have the two
URLs? This is how our host provider set us up, rather than using a
redirect to abuseneglect.com - as far as I understand, this is how
they handle having two domain names pointing to the same site. I have
been submitting the longer URL to the engines, as it is the one with
titles, keywords, etc. However, when I tell a client to visit our
site at www.abuseneglect.com, they don't see any titles, just the URL
as a title on each page. I understand this is because it is a mirror
site, and I guess there's no way around that?

Second, what would happen if we decided to independently host each URL
but kept the content the same on each - would this get us black-listed
from Google? Basically, we want the work we have already done with
SEO on http://www.nursing-home-abuse-neglec...nformation.com to be
worthwhile, but we also want to make www.abuseneglect.com useful and
worthwhile as well. But perhaps it would just be double the work to
host each? And would that even be legal/accepted, as I mentioned
above?

Third, I am wondering about incoming links. I have been writing to
comparable sites asking for a link to us at www.abuseneglect.com, and
I have had a few people link to us already. However, will Google see
this if it is looking for links to
http://www.nursing-home-abuse-neglec...nformation.com, or will it think
nobody is linking to that site because all the links are actually
going to www.abuseneglect.com? I know link popularity plays a big
part in determining Google ranking, and I don't want us to be hurt by
having the two URLs.

Sorry about the length of the post - I have lots of questions. Hope
someone can help.