I want to register a domain, and register an email with that domain.

If you ever look up an LLC through the Secretary of State website, your name will show up because it's in the public domain.

Also, a big manufacturing plant was supposed to move in too and bring hundreds of jobs, and that just fell through at the last minute (they had already won all of the public domain suits to take properties for the building, and had broken ground). 

Do you recommend me using my first and last name as my domain name for the site or create a company name for it?

For example, if I own a property that the city is trying to take in eminent domain, I am not allowed to testify on value, because I'm not an expert, even though I may have been investing for 30 years and know what I'm speaking of.

I used to have one with them, but couldn't use my domain there and for my email, so had to switch to go daddy for website. 

Recently I was approached by someone (who I know well) about investing with me, and he wants us to set up a separate LLC to do mainly buy and hold rentals as well as do some flips.He wants to provide 100% of the funding, but essentially be a silent partner, and do none of the work (he has another business that he works at full time).

to see if it comes up or if anyone owns the domain.

If the owner does ask, I think it would be fair to just mention it was found through public domain?

You can create a custom e-mail address through Google for something like $6/month once you purchase the Domain.