Wanted to contribute a bit since I got great answers to my questions.
For those wondering, it is possible to walk to the Southampton cruise terminal if you have an hotel nearby. We did it in June. Obviously only recommended for fit travelers, especially if it is hot. The taxi ride is cheap and everyone will recommend it, but if you want to walk, you can. To the Mayflower Terminal, google maps gives a pretty good itinerary, through Dock Gate 10, W Bay Rd, but you don't have to zig zag at the end, instead use Imperial Way.
We stayed at the AllAccord Novotel, basic rooms, but ok for a night. Other hotels nearby. We walked from the train station. They were rebuilding the access to the train station but it was a very easy walk. There are a few simple restaurants nearby, we also walked to the SeaCity Museum and did the Stonehenge Stone Circle Experience via a prebooked transfer with Paragon Taxi, exceptional service and on time.