A pushover is a person who is easy to convince and a submissive person is a person who obeys commands. Pushover は簡単に納得する人で、submissive person は命令に従う人です。 ちょっと同じぐらい。 (すみません、日本語がまだまだです。)

Pushovers are people who let people 'walk over' and take advantage of them. A submissive person can just be someone who is quiet, likes other people to take the lead and doesn't like to be the centre of attention.

I believe the terms mean the same thing, but are used in different situations. "Submissive" is a more formal term, while "pushover" is more informal. Submissive is used to describe a person who happily obeys commands without question. This is generally used when a person is expected to be submissive. For example, a child obeying their parents, a wife obeying her husband, or a student obeying their teacher. "Pushover" is also more often used as an insult. It is often used to describe a leader, or someone in a position of power that is acting submissive when they should not be. A pushover may be a teacher who allows their students to forget their homework without consequences. A parent who allows their child to stay out past curfew with no punishment could be called a pushover. I would say that "submissive" is generally less common in every day conversation though. Finally, I would just add a quick edit to your sentences above: 1. She is submissive and does anything her friends ask HER to do. 2. She is a pushover and does anything her friends ask HER to do. 3. She does anything her friends ask her to do. - This is probably the best way to get your point across, as the other two sentences are redundant. But, if you'd like to pick one, I'd say #2 sounds more natural.

"Submissive" is less correct. It is more of a term from certain adult movies.