Yes, you must prime all MDF surfaces.  Prime the wainscoting and moldings with a good acrylic primer. We have had great success with Fresh Start Primer (high hiding) from Benjamin Moore because it covers well, dries fast and sands easily.

Old timers will tell you that Oil Paints is better than Waterborne Paints because it is more durable and will lay better (less brush lines). Paint Technology has changes dramtically in recent years and we believe the benefits of Waterborne Alkyd Paints out weigh the benefits of Oil Paints and we recommend using them. There are 3 cons to Oil Paints - they yellow over time, have high VOCs (Smell), and the clean up requires harsh chemicals. A good quality Latex Trim Paint will not yellow, is durable and will self level giving a great apperance that will last.

There are a number of good quality paints on the market.  Our favorite at this time is Benjamin Moore's Advance Interior Paint.  It is a premium quality Water Reducible Alkyd that is low VOC.  We like this paint because it covers well, does not drag and levels very well (minimizes brush marks). You can go with a satin or a semi-gloss paint sheen. The satin sheen will be more forgiving and will hide most imperfections from your nail heads and seams.

This all depends on the type of paint that is used.  If the paint is thick it may leave brush drag marks so you may need to add a paint additive.  We recommend using a good quality primer and paint so usually you will not be required to thin it out.

This really is a personal preference. If you are going to spray paint the panels Semi-Gloss is a good choice. If you are going to brush paint the panels then we recommend a Satin Finish. The Semi-Gloss Finish will show imperfections such as filled nail holes and seams if they where not prepped well but the Satin Finish will be more forgiving.