Asking does anyone run into the same problem? is also grammatical, but has no sense in this context when applied after a simple past (... when I installed ...).
In other contexts it would just mean a different thing: you would be asking if anyone runs into the same problem on a regular basis, or is running into the same problem in the present, like if they are having the same problem as you right now.
However, with has anyone run into the same problem? you would be asking if someone has already (at least once, but in the past) run with the same problem, and would definitely make sense because it is compatible with the simple past used in the previous sentences.
"Will" implies something is going to happen in the future (in this case, getting your visa). But it's unnecessary in the first sentence: "how soon" provides enough context that you don't need the word "will." Like rudra, I'd interpret both sentences as conveying the same meaning. In fact, I might also suggest:
It depends on when I get my visa.
as that would essentially mean the same thing, too.
However, if you were asking the question:
"How soon will I get my visa?"
then the "will" is necessary. (You wouldn't say, "How soon I get my visa?")