You can travel on your fiance visa as long as you are a fiance. Once you are married you become a spouse, and your fiance visa isn't valid for reentry anymore. If you can afford an in-person FLR(M) application, you can schedule that for right after your wedding, then you may need to wait a week or so to get your ID card before you can travel. If you want to do the FLR(M) by post, you'll have to postpone your honeymoon or else go somewhere in the UK.
In order to use the NHS, you will need to register with your local GP. It shouldn't be a difficult process, you just fill out a form with your name and address and answer some questions about your medical history. They will enter you into their system and give you your NHS number. If the GP tells you that you cannot register right away or anything along those lines DO NOT BELIEVE IT! You are eligible for NHS as soon as you arrive, and can begin to use it immediately. My GP was even nice enough to let me have an appointment the day after I registered, even though their computers were down, so I wasn't officially in the system. I'd also recommend not going to the GP alone. My MIL came with me when I registered, and I think things might have gone more smoothly because she knew the people there.
As for the passport, if it were me, I'd keep it in a safe place alongside my own passport. There's too much risk of checked bags being lost, IMO, to put it there. If they question you about it, tell them the truth, and have your fiance's phone number on hand in case they want to call him to ask about it.
