First off, I believe that the only passports that the ship takes from passengers are those who are non-US citizens, but I may be wrong. I know that when we cruised, we didn't have passports but the only ones that were kept at the pursers desk or wherever were those who are either Canadians or Europeans or other non-US citizens.
Next, yes you will need to bring a photo ID on land with you and your ships key card. There were sometimes when the ships card was enough, but sometimes they wanted to see a photo ID. If you want to do any shopping or pay for excursions on shore then you will need to bring money or credit card with you. The only time you can charge on your key card is the private Island of Great Stirrup Cay and since you are going to Belize, it sounds like this is a Western Caribbean cruise and you will not be going to the private Island. It is clear on the East near the Bahamas.
There will be excursions to sign up for at the docks and chances are your ship will have a port talk with a port consultant on board. I strongly reccommend going to the port talks if the ship has them. They will tell you about the best shopping, prices of taxis if touring on your own and other beaches or things that can be done if you don't want ship excursions. He or she might be able to help you answer questions you might have.
Tenders go all day, but depending on the Island, you may not want to tender anymore than you have to. Some of the tender rides are only about 10 minutes, like Grand Cayman but in Belize the tender takes 30-45 minutes and that would be a waste of good time tendering too much. In Grand Cayman, we did our shopping and sightseeing first then tendered back to the ship to leave money and cameras there and then went snorkeling. Some places especially if you are on an excursion may have safe areas to put valuables. Hope this helps and answers your questions.