Can me and my Gf be engaged officially without a blessing from parents?

In my culture that would be completely acceptable. But I don't know where you live or what the social rules are there.

If you mutually decide to be engaged, it has nothing to do with your parents. It's between you and the gf. If they don't like it, i guess they will have to learn to accept it.

Depending on your religion, culture, and age, most people can with the exception of the main barriers. So absolutely unless you feel like you cannot due to your faith or if you are underage.

it depends on how old the both of you are

Only if she is actually old enough to not require parental permission to marry.

If you are adults,  Yes, you have no obligation for blessings from your parents.  If they persist, then not let them visit the kids,  when you have them. 

Well yeah, there's no law, but I would encourage you to keep your parents included and the door open. Marriages last longer with family support.