Anything else that I need to take care of? Thank you in advance.

Send what you have with the cover letter from either one of you.

Can I travel to non-contiguous states without having a decision on my asylum case?

With proper valid ID you can travel to any of the United States, subject to inspection by TSA.

Advance Parole I-512- travel

If the passport is still valid, with your green card you can travel.

Can I still get married to a US citizen, here in US, while my marriage in the Philippines isn't annulled yet, what to do?

While a state issue, as far as I know there are no jurisdictions in the US that permit a legal marriage unless you are single. You are not single.

One cover letter for all the forms in one package is okay.

I 864 Federal Income Tax Return Information , 2014 was enough , but 2013 2012?

You only need to provide 2014 with the I-864.

What is the proper way to help him become instate as soon as possible,

I think you meant to ask about refugee status and not asylum. Either way in state tuition is up to the schools and/or state law.

Request for Initial Evidence on Affidavit of Support I-864, Adjusting my Status

Your best bet is to get a cosponsor or wait until you have all the necessary documentation.

How can I get my boyfriend back into the country?

There is a lot more information needed so you would do well to visit with a local immigration attorney. In the meantime look up what is necessary...

Does not matter as long as they can read it.