1. If staying on paved roads, not exploring rural routes or crossing over to Caribbean side in CA, then road bikes are OK but not ideal (for many reasons suited for another thread). Tires alone don't make a bike suited for off road or even rough, muddy dirt roads. Yes, it has and can be done, but just depends which bike you choose.

True, plenty of beat on street bikes cheap (you get what you pay for).
Dual sports can be more expensive, but deals can be found with patience and luck. Buying a worn out bike can be risky unless you have exemplary mechanical abilities and a parts budget.

I assume, after selling off bikes in Panama you plan to travel by Bus once in S. America? I would much rather have the bike in S. America than Central. More to explore, greater distances, more diversity of terrain, culture et al. The good stuff!

So rather than sell off bike in Panama (hard to do) maybe consider either shipping bikes to Colombia (about $2000 each for bike and you).

OR ... use public transport though out USA, Mexico and CA, then buy a bike in Colombia. Many posted for sale everyday here on HUBB. Also consider the many Chinese 250cc bikes for sale all through S. America. Cheap and getting better. Or start in Colombia and do CA/Mex/USA on way North?

Question: have either of you ever ridden motorcycles? Training?

2. By the finish of Mexico portion your research and personal contacts will give you tips for sights and must see spots in CA. Reading ADV Rider Ride reports will also help flush out your plans. Any good guide book would be a start ... even an older one.

3. and 4. See above.