Can "visiting" be used here instead of "having visited"?

Short answer: yes. There's some very small nuance. But in general they both work here

Do not make any payments in advance before visiting the flat. Do not make any payments in advance before having visited the flat. There a tiny nuance in meaning between the two, but I'm not sure if the difference changes the pragmatics of the sentence. I would say yes, you could change the words.

On other day i've watched video about "having V(ed)" on one of EngVid YouTube channels. if I remember that correctly, it should be seem as "after have had V(ed)" ,or smth like that. I would say like, gerund will be fine there, but sentence emphasize the perfected nature of action, that you definitely already checked the flat BEFORE the payment.

I’m a native English speaker and if this was a multiple choice test, I would pick visiting. lol

Yes, visiting makes it clearer.

"Having visited" is better. "visiting" allows an option for making a payment WHILE visiting. The author is suggesting that payments be made after a visit.

Before you have visited is more appropriate than having visited. I would only use "having visited" at the beginning of a sentence, not in the middle. Having visited the flat, I decided I didn't want to live there. That said, in regards to real estate I would substitute visit/visited with inspect/inspected.

I think visiting flows better but they both work in this case. There could be a difference if you talk about things you regularly do. Like "I packed my swimsuit before visiting the beach" is a basic set of steps. "I packed my swimsuit before having visited the beach" means you never went to that beach before, maybe you would have done something different. Maybe it's a nude beach. Idk.

I still wouldn’t use western union even if I had visited the flat

Yes, and while people can debate style I would strongly prefer "visiting". It's clearer and more direct. It doesn't actually change the meaning in any way that matters, since the point of the sentence is to make sure you don't pay for a flat you've never seen. "Before visiting" might imply it's OK to pay \*during\* the visit as opposed to \*after\*, but so what? That would also mitigate the fraud.