Fuse/breaker size is primarily determined by the safe current carrying capacity of the wire after it. They're there to prevent fire in the event of some catastrophic problem (like a short to ground).
For the 10 AWG solar controller wire, 30A would be reasonable, as close to the battery as practical. From a voltage drop perspective, a heavier wire may be beneficial here.
For the 4 AWG wire to the inverter, 100A might be appropriate, and probably a slow-blow fuse or thermal curve circuit breaker would be most appropriate. This wire seems rather marginally sized for a full 1000W load from the inverter.
For a catastrophic fuse at the battery, that depends on the details of what is connected and how it's connected and what your maximum possible current draw is. On my motorhome I believe it came with a 175A fuse there (and I think 2 gauge wiring for the battery post connections). The only time such a high current could possibly be expected is when starting the generator or the main engine with the emergency start switch pressed.
On a trailer, my general understanding is that the connection to the breakaway switch (and thence to the trailer brakes) should not be fused. I do not claim to be an expert, nor did I sleep at Holiday Inn last night.
