Can I use different kinds of doors in different areas of the house?

You might get away with it if you can't see the bedroom doors from the main living area.

While it is possible for MDF to be stained, the final outcome on what the 'finish' will look like may not meet with your expectations. The reason is that MDF does not have any grain to it, as it is not real wood.

Also as MDF does not absorb stain to the same extent as real wood, a polyurethane coat over the stain is advisable.

My suggestion if you really want the doors to look the same as the living room windows, is to purchase unfinished genuine wood doors, then either diy or hire out to have them stained.

When I paint clients doors, I take them down and do them on saw horses in the garage. Just use a large screw driver on the hinge and tap with a hammer to lift each out.

If you are going to stain yourself, I would use wood conditioner first to get an even finish. I like to pour the stain on a rag and quickly apply.

If you want to make the MDF doors look like wood, paint them first with the lighter of the 2 wood tones, then apply the darker tone gel stain over the dry paint and run a wood graining tool over it..... or here is another technique..... https://www.youtube.com/watch?v=liQuCa1pyWA

I think that would be totally fine. You could pay to have the living room, etc ones done but do the bedroom doors yourself.