The last 3 places I have rented since 2019 I paid the 2 months balance at time contract was signed and handed over. I had paid 1 month rent to reserve the property as all three were in demand. The one I am in now I did not actually view as so many after it. If you pay the deposit to reserve then yes the agent should give you an immediate receipt.
Hope it goes well for you this time. Let us know. One of my very good friends who I used to work with had a "Difficult" passport. Needed visas for everywhere including Thailand. He spoke at many medical conferences globally as he was senior in his field. Now he took Singapore Citizenship. Can travel almost anywhere visa free.
As for DVT and nationality it is 100% true certain nationalities and locations are scrutinised more than others. Officials have pretty much said it without saying it directly. Same as in other countries like Vietnam. Certain passport holders very much restricted without being a published list.