Yes you can submit the I485 without the SSN. You don't need a state ID either. There's a question on the 485 and 765 if you want a social security number issued when your form is approved.
Please complete your timelines. Not sure when you entered the US but SSA will not issue a number beyond the 75 day mark of the I 94. Others may have more experience with the Vietnamese naming convention. For the I485 list the desired name first. Any other names should be listed below the first line on form.
As for state ID, you have to wait until the employment authorization is received. Definitely spend the extra money for it.
File the I485 as soon as possible and carry the receipt notification as proof of Status
Each entry is at the discretion of cbp. How is she able to afford to travel with such frequency? What ties does she have to return to her country at the end of the authorized stay?
I think you're getting too confident coming on a visit visa. Have your significant other visit you in Europe a few more times or meet in 3rd country before filing for k1 or cr1