Hi Sinochief, good questions.
1. Wet gear
You really want to wash the salt and any sand off as soon as practical. It might be an idea to carry water for this purpose but I don't currently.
Car Storage
I usually have a large, unsealed plastic dive box in the back of my car. Most gear goes in there. My wetsuit, gloves, socks go in a big Ikea bag on top (I have a second one for family wetsuits). I have a tough, stiff shopping bag with plastic-reinforced base, as reinforcement - I put my weight belt and weight vest in that. I store my spearguns in a large rod bag, alongside fishing rods.
Flush & Dry:
When I get home or back to where I am staying, I unpack it all, rinse it all (pref. in a large Belfast sink, bath or shower) and leave it somewhere to dry. BBQ areas tend to be good for this: put small things on the table/chairs, draper wetsuits over the back of chairs or hang them under the umbrella (which also offers some protection from sun/rain). Washing lines can be handy but dive gear is very heavy. You can sometimes drip dry on racks or in shower in doors - it rains a lot here! I repack stuff once it is dry. Spearguns, I usually prop up against something and often invert them once part dry, to allow any trapped water to escape & so reduce salt/grit retention.
2. Storing fish
In the sea:
UK waters are generally not sharky, so when I spear a fish, I put it on a cable stringer. My main fish stringer is attached to my float. I often wear a cuttable, nylon waist stringer. Stringer keep the fish cool and clean and can act as a visual & scent attractor to some other fish. If you have sharky waters, I believe some use "planche"("plank") floats - essentially their dive floats are floating fish storage boxes [search this forum for more info. on these, there are some neat images]. Talk to the locals to find out what is appropriate where you are. I recommend that beginners avoid waist stringers.
Back on land:
I have used both approaches that you describe successfully, so up to you. If driving to a dive, I take an ice box with cold ice packs - and leave some shopping bags and large plastic bags in the car in case I forget the box or catch something that won't fit the box. I used to gut the fish at home, before flushing my gear [see above], so the fish could be frozen or ready for the table a.s.a.p. I retain fish guts & heads and freeze them for lobster pot bait or chum/burley, or, sometimes, my wife will use the head & bones to make fish stock (good base for seafood soups).
But recently, I have tended to gut fish at sea, immediately after dispatching them. It reduces the post dive workload, when many things need to be done quickly. It means less mess to clean up back home. It creates instant chum, right where you are fishing now and means less mass and volume to tow around with you. You still have the fish heads, which can be used in lobster pots or for stock. The gutted fish will of course look smaller and weigh less, if weights and photographs are important to you, post dive.
