Another instructor is pushing me out of the classroom right after my class ends
You should first find out what is the standard convention for when you have to vacate the room.  Since the other instructor is in your department, you can take this up at the department level: talk to your advisor and/or a trusted faculty mentor about it.  

In my experience though this is often left a little fungible and people need to be reasonable about it.  In my opinion it is reasonable to expect to have access to the room five minutes before your class begins.  Ideally the previous instructor will have vacated the room at that point; if not, s/he should be occupying the room in a way which doesn't interfere with your own use of it.  (For instance, a lot of times students will not enter a room if an instructor is still writing on the blackboard.  It can be annoying to come to your class a few minutes before you want to start and find everyone waiting out in the hall.)

In the absence of a clear directive about how to split the time, it seems reasonable for the departing instructor to take half of the time, as long as they leave at least five minutes for the other instructor.  In your case 20 minutes is a very nice cushion (in my university it's 15 minutes), to the extent that I find it a little strange that the other instructor is arriving 15-20 minutes early.  What is she doing with that time?  Do more than a few students arrive that early?  If you are using the time to talk to your students and she's using the time for nothing, then I think you are morally in the right and should push back on it a bit.  You should be able to stay for the first half of the 20 minute break.

How should you do that?  As above, I would find out what other people do.  If the standard convention is that you can take those 10 minutes, then talk to the department head (or other departmental authority figure; e.g. whoever schedules the teaching) about it and get their support.  (You don't need to phrase it as a grievance or even identify the other instructor by name.)  Assuming you get backed, then you should pay a visit to the instructor in her office and explain that you looked into the matter, that you have a clear instructional purpose for how to use the time, and that the head (or whoever) has confirmed that this is a standard practice.  Definitely do this outside of the classroom so that you two can have the conversation you need to have not in front of the students and so that there is not a direct confrontation to be resolved.

In my opinion it is probably not worth trying to seek redress for your poor treatment by this instructor.  We are not obligated to be nice to each other; it's just better if we are.  If you respond to the treatment in a professional way and show that you are neither going to get pushed around nor retaliate in any personal or emotional way, then you're basically just rising above.  What some random person who you're only going to see for one more semester thinks of you isn't really so important, is it?  Especially if you're confident that you're in the right?