You have a very common issue in RVs. I don't understand why manufacturers do this - even in very expensive units. Obviously designed by men with no kitchen experience.... They could often re-jig layouts to make a little a little more surface area. One feature that sold us on our TT is the rear kitchen. In comparison to many counters, our surface area is huge.
1. Yes on the folding extension to right of sink. Go buy an inexpensive wood cutting board somewhere. Lee Valley has some nice folding brackets which are very strong:
Lee Valley brackets Nape & Vogt also has a similar one:
Nape & Vogt bracketOn this folding extension, it was a very tight squeeze into the dinette seat, so I could not make the surfaces the same height.
2. Making a cutout for the sink is a good idea. Use HDPE plastic and not wood. I made a wood one once and even though it was well coated with mineral oil, the constant water on it caused it to warp and cup. HDPE is easy to cut with a jigsaw and fine tooth metal blade and sanded with ordinary sandpaper. Drill a 1" finger hole near the perimeter. HDPE is readily available at plastics shops in various thicknesses.
3. Get or make a wood cutting board large enough to go over the entire cooktop to the rear wall and side wall. They are available at Country Mouldings:
Country Mouldings wood countertops They aren't cheap. I bought a 24"x24"x1" maple one from Country Mouldings last winter and it looks like the price has gone up quite a bit. Easy to make if you can fine some good edge grain maple or other species. You'll need to raise it an inch or so from the counter surface to clear the grate. Be aware that RV countertops are often warped so you will need to take that into account for whatever method you use to support the countertop.
If your kitchen is relatively centered over the axles, you will find that you can leave a wood cutting board over the cooktop and a cutout in the sink without having them bounce onto the floor while in transit. Our first TT was 20' with the kitchen over the axles and the large cutting board I made for the cooktop and the sink cutout stayed firmly in place while travelling. A rear kitchen is a different matter.
If you need a receptacle for the newly acquired counter space, you can hide romex under the counter top and run to the nearest recept. You can also mount a recept. to the underside of an upper cabinet.
I note that you have a smaller TT. Every square inch of storage space is important in a smaller TT. I note that there is a blank panel on an opening under the sink. If it were me, I would see if you can get a stock cabinet door from the factory to cover the opening and have either a hinged door or make a drawer. For a drawer, it would probably have to be only partial depth and be L-shaped to clear the waste pipe. You can relocate waste and water piping if needed which is relatively easy to do. You could get a custom door made at a cabinet shop but it would cost more than a stock door.