Spinning head!

Thinking about technical upgrades for Waddles is making my head hurt.  This is where I am the weakest in knowledge and confidence.    My trailer wiring needs to have a 7-pin plug in order for the trailer battery to charge while we are driving.    I don't have that, and need to get it changed.   Not sure where to start, but will contact someone who installs hitches to see what my options are!  These plugs are on the trailer:

This plug is on the car:

I have been reading about recommendations to upgrade the battery to one that holds a charge for much longer than many.   I have no idea if what I have needs to be replaced, or if it is good as is!

I have a plug for plugging in a solar panel, which would allow me to 'boondock' (camp off the grid) and keep my battery charged.  I'm hearing discussions that I need a converter to go with a panel to avoid over charging.   I'm not sure where to proceed with this, and I'd love to camp mostly off the grid!

And Waddles comes with a cooler, which runs off the battery.   I'm reading about non battery coolers that are super duper and will keep things cold for like 5 days.   Expensive, but I think would be better than running the battery down.

I have bought replacement fuses to have on hand, just because!  This one was easy.

I need expert advice!