1/ Forget about a job until you have your residency. Then it will still be difficult and the pay is very low. But you never know what will come up.
2/ Banks are banks, so pretty much the same. The main banks are, Belize Bank, Atlantic Bank and Scotia Bank (Canadian) All of them take ATM cards for withdrawals and deposits. All accounts at these banks will be in Belize dollars, not USD.
3/ The housing market depends on "location"!! Lots of homes for sale as well as land. Homes can be had very cheap or expensive, naturally the waterfront properties are the most expensive. Land is for sale every where, jungle land from $1000.00 BZD per acre to waterfront at from $100,000.00 USD and up, depending on location.
4/ Cost of living depends on how you want to live and where you want to live. Living in a touristy place like San Pedro will be at least 50% more than living in Corozal. I know of couples that live a simple life in Belize on less than $800.00 USD/month and I'm sure many live even cheaper.
5 /You will not be able to get a mortgage on a house in Belize as new arrivals, no jobs, no history in Belize and what kind of assets will you have? Then again the interest rates in Belize are very high compared to Canada or the USA, you'd be looking at somewhere around 10% interest if you could get a loan.
6/ You must stay in the country for 1 year to get your residency. Once you have this I believe it is 3 years later you can apply for citizenship. Dual Citizenship is no issue.
7/ Transportation, yes there is, but it is not public, but an array private buses. They go everywhere and are very cheap. Walking distance is dependent, on where you live. If you live close to the town or village you decide to live then go ahead and walk. Bicycles are the main mode of transportation. Cars and gas are very expensive for the average person in Belize.
8/ Shops, hate to keep saying it, but again it depends on where you live. A small village will have limited selection, Belize City at Brodie's, you can get pretty much anything you need, but not always what you want. Anything made in Belize is usually very reasonably priced, imported goods, expensive.
We live in the North of Belize close to the Mexican boarder, so we head off to Chetumal where you can get absolutely anything you need and want. They have all the usual stores, including Sam's club, Home Depot as well as all the fast food you dream of, getting it back into Belize is a whole other story.
Hope I answer a few of your questions. Once you have visited Belize and traveled around, you'll be in a position to ask better questions and get more exact answers.