You really need to consult a competent real estate attorney in Florida. There are so many complications here! Here is my opinion, completely uninformed by any legal education.
If your father does not wish to co-own the house, he can give you the 50% deposit as a gift. Then you can use it as a down payment. However, if he is a US citizen, he will have to pay a federal gift tax. If you are both non-US citizens, he can give you money outside the US without any US gift tax -- but who knows what the Swedish rules are on gifts?
Or he can give you a loan for the 50% -- but it better not be secured on the property or the bank will not give you a mortgage. You will have to declare this loan when you apply for a mortgage and it might affect your ability to get a loan.
If you own the house 100% yourself, what you do with the rental income is your own choice. However, it's income to you and you alone pay taxes on it If your dad gave you money as a gift, you are certainly free to give him money from the rental income back, but you might have to pay US gift tax when you give him money. If he gave you a loan, you can repay the loan without any gift tax consequences. If he is charging you interest, you might be able to deduct the cost of the interest on your taxes, but be aware that interest paid between close relatives is scrutinized by the IRS very carefully.
If your father agrees to co-own the house, he can certainly pay the 50% downpayment. The mortgage, however will always be in the names of both the owners -- no bank will give you a mortgage for half a house. The income and expenses for the house can be divided up based on whatever percentage ownership each of you has -- and this % need not be related to the actual cash investment each of you made in the house because perhaps you might be putting in some extra sweat equity. This is really the cleanest solution here. And do make sure that the join ownership is "right of survivorship" so that should he die before you, you automatically get the entire house without it going through your father's estate.
LLC: There is another discussion on this community of the plusses and minuses of using an LLC to own a vacation rental. Search for "LLC" to find it. Most people form an LLC to own the house to limit liability in case a guest is injured at your house, or maybe causes a fire that damages nearby homes. Theoretically, when an LLC owns a house, the investors in the LLC are protected against lawsuits that go after their personal assets. My impression is that it might be very difficult to get a loan since the LLC has no other assets or income to base a loan on. But I've never heard of someone forming an LLC just to collect the rentals. I can't imagine this is even possible.
Please, talk to a lawyer!!!
These are some huge legal questions. Not only do you need to speak with an attorney but an accountant, real estate professional and a tax professional as well. Please don't take ANYTHING you read on the boards as legal advice as we are all just vacation rental owners and renters giving an opinion.