Rental car pick up and drop off

Portsmouth (like Dover) is a sizeable city and a channel ferry port, so there are likely car rental offices parallel to whatever company you use out of Dover. However, you could reduce your train fare by dropping the car at Gatwick. From Gatwick you'll have a choice of public transport to central London wherever your hotel is. https://www.gatwickairport.com/to-and-from/gatwick-to-london/

I would suggest not turning in the car at Gatwick, it’s a 4 hour drive from Plymouth. Heathrow is closer, but it’s a 3 1/2 hour drive. I just looked on nationalrail.co.uk, if you buy your your train ticket in advance you can get a ticket for £21.50 each person that price is for 11 weeks in advance. Then I did a 7 weeks in advance there were tickets for £21.50 and £28.00, the train ride is 3 hours 26 minutes for a 9:49am train. The train isn’t any faster (then driving to Heathrow, but faster then going to Gatwick., but it would be more relaxing. You never know what kind of traffic you might run into, so it might take longer then you think driving to one of the airports to turn in the car. IMO I think taking a train is a wiser choice.

I don't see any problem with your plan. The biggest problem will be getting from the boat to the car rental place. Have you got that organised? Maybe taxi?

Trundling along the south coast from Rye to Portsmouth should be OK - you'll be stopping along the way at various places?

It is certainly straightforward to go by train from Rye to Portsmouth by train, and it would be easier on the driver, but there is so much to see along the south coast if you take enough time going by car is fine. I could do it in a day but I'd much prefer to take at least 2 days.

What sort of thing do you want to stop at?

Portsmouth to London Waterloo or London Victoria (different prices and speed and route) is straightforward enough.

Unfortunately there are no car rental firms located in or near to the ferry terminal. Most of the rental firms are located in the north of the City and are unhelpfully not located close to public transport. The exception to this is Europcar which is situated close to Portsmouth Harbour from where you can take the train to London.

Enterprise are based in two locations in Portsmouth, both within industrial estates however they will take you to a location of your choice once you've returned the car. They're both within easy reach from major access roads into Portsmouth, you just might need a little guidance from a sat nav to locate them within the industrial estate.

If you're planning on staying in Portsmouth then I'd probably opt for an Enterprise rental. They are well serviced in the UK and with their option to take you to your final destination it is an ideal way of getting where you want to be (or take a taxi from the rental place, it's a small island so the fare will be within the £5 mark if you order a minicab rather than flag one down, either Uber or the main taxi firm Aquacars).

Portsmouth is an easily walkable city and you can take the train to London from any of the three major stations as they are all on the same line. The trains start from Portsmouth Harbour, the next station is Portsmouth and Southsea followed by Fratton. Your choice will most likely be dictated by location and ease of access but any one of them will do. Of all of them, Fratton is the less salubrious location and does feel a bit drudgy and intimidating being located in a poorer part of the city and where you're most likely to encounter some of Portsmouth's more 'interesting' characters. I prefer the newly rebuilt Portsmouth Harbour where overlooking the harbour and HMS Warrior is better than Fratton's Lidl and hand car wash.