How do i sign up for the USPTO Trademark without being based in the US?

Good answers above. There had been too much fraud and error over the years coming directly from non-US applicants so requiring a US lawyer to weed...

You will need to prove you have trademark rights -- (1) you've used it as a trademark, in commerce, it's distinctive and enforceable (2) the...

If my business owns the trademakr for aquadream, can I also use aqua dream, or AquaDream or even Aqua Dream or Aquadream?

Your question is pretty unclear. If you use "aquadream" as a trademark for particular goods, or a service mark for particular services, you could...

Probably trademark or copyright infringement not patent. Retain a lawyer to see if you have any defenses -- selling counterfeits has significant...

Assumed name?

It's unlikely there's any crime just by not registering an assumed name, if that's all that happened. If it were part of something else that would...

The above answers are good ones. I also wonder how you could be "in a completely different industry" if you are "promoting our golf products."

Probably, but it depends on the logo. If there are some features like interlocking letters the logo rather than plain block letters might be...

Since you must be "using" your mark on all the goods you list in your application, before registration issues, it makes no sense to have a...

Good answer above. You either own or have a lease to the store. You define who can do what on your real estate. It might be good form to post a...

To best protect the words of a trademark, is the logo important? I want the word(s) itself protected, might change the logo.

Balancing these things are one of the places where advice from an experienced trademark lawyer is valuable. Plain block letters could be more...