Welcome to the forum Todd.
Others will likely have some good advice but here are my thoughts to get started:
Re: #1 - there are several good battery brands out there. My suggestion is to decide on the charging system first then select the battery or even better the batteries
Not much point putting in a great battery with a substandard charging system.
#2 - the Onan generates 110V AC. An automatic transfer switch kicks in when you start the generator allowing 110V AC to pass through to the coach. The converter/charger in the coach converts 110V AC to 12V (nominal) DC. It is unlikely that your chassis battery gets charged by the converter/charger in the coach unless the setup was modified by a previous owner. Some newer coaches have that feature. My older modified van has that feature.
#3 - there are several excellent solar installation projects with photos and details on this forum. My suggestion here is to size the solar based on the size of the battery bank or anticipated size of the battery bank. 200W solar with a single battery might be too much for example. Maybe somebody here knows of a good ratio of solar wattage to battery amp hours to use as a guide.