With the shoulder press... do you bring the dumbells down all of the way? or stop when your arms are parallel to the floor? Why?
Bring the dumbbells down to your shoulders. (touch your shoulders) A full range of motion is a good plan. If you get into the habit of stopping at some arbitrary point you will end up doing shorter and shorter range of motion lifts as you are pushing yourself and using more weight.
With the seated row, do you do a row at all? As in move your entire self?
You can. Leave your legs in a fixed position. But if you get some forward and backward movement at the hip it is a really good exercise. This will allow you to use heavier weight, get a good reach at the "bottom" of the lift for a good range of motion.
Do not lean too far back. Leaning forward some and then bringing yourself back to an upright/just past upright position is a good route to go.
When I do seated rows I usually do them with some movement at the hip. I feel it is a more complete back exercise when done this way.
Or do you just basically keep your back completely straight and just move the weight with only arms? WhY?
You can do it this way to. Using a more strict style can help you learn better how to protract and retract your shoulders and get more range of motion out of your upper back. Especially if this is something you have trouble with.
Rowing in general is not just moving your arms. It is reaching your shoulders forward (protraction) and then pulling them back. (retraction) Learning to do this will help build your upper and middle back a lot more.
Think of initiating the row at the shoulder. The first move is pulling your shoulders back and puffing your chest up. That will recruit your back first and you will be able to pull with your back more through the entire range of motion. The second movement is pulling your elbows back.
A lot of people initiate rowing exercises at the elbow, starting the exercise with their biceps. This really decreases the amount of weight that can be used as well as decreases the overall activation of the muscles in your back.