Molded door so no true floating panels, therefor caulking will not be necessary. I would select the same paint/finish for the doors as used for the jamb and trim. That is personal, for appearance, not a necessity. I use a semi-gloss enamel. Not familiar with the paint you reference. Make certain the surfaces are clean and free of any oil. Wash with Dirtex, or similar, if necessary. You might find that a foam roller will give you a smooth finish but you'll still need a brush for the 6-panel inside edges and the edges of the door itself. A good quality paint will self-level, just don't rework following application. Yes, personally I prefer to paint doors flat, resting on horses as I have a better chance of eliminating runs. I do two light coats a day apart, flip the door and do the reverse side. I remove the hinges and knobs/mechanism too - less trimming. I'm sure one of the pros can offer some techniques but I don't paint doors often so the above has served me well. Hope this helps, enjoy the new bathroom.

Any solid door is more sound transfer resistant than a hollow core door and the additional thickness of a raised panel wood door is offset by the greater density and tighter joints of a flat pane MDF door and its doubtful that a raised panel MDF door is much better than an MDF flat panel door. To make these doors more sound transfer resistant requires the installation of a perimeter gasketing system.

Any time, effort and money spent on modifying these is a waste. It will look bad. Just leave them until you can afford to replace them with whatever you prefer. Is the rest of your house modern? Do these stand out as out of place? I know you personally hate them, but they are very common and nobody else would probably notice them unless it's a glaring contradiction in styles with the rest of your house. Could you afford to replace just the doors in the public areas of your house and leave the private area doors for another time?

We also bought a house a couple years ago that have these doors.... but they are painted white. If yours were solid wood I would not paint but since they aren't, painting will update the look ... a little. I think you need to live in the home for awhile before committing to changes that cost a lot. My concern is that if you change to the shaker doors you like you would also have to change all the door trim and floor molding to a simpler look to match. The cost could snowball especially if you can't diy and have to pay someone to do everything. Even painting is a major cost if you aren't doing it yourself. Is it your forever home? It could be worth your time and money to change if you plan on staying here awhile. But if it is a long time home you also have time to make changes in stages you can prioritize. Our home IS our forever home. I'm not crazy about a lot of things but changing the doors are far down the list of things I want to change. First is kitchen appliances, changing flooring and hopefully someday changing our master closet/bath. I hate the layout but at this point don't know how to make it better. Because of that it is the only place in our house that hasn't been repainted. But my husband tackles a lot of our home projects. Time and money.