The town starts out with the strangeness at a minimum, you'll see small strange plants in the neighbourhood and a few infected sims running around. Even your welcome wagon will consist of infected who will place a strange fruit in your fridge, which will then be confiscated by sims in black. If you don't want that to happen, just ignore the welcome wagon. Other than that, you'll meet scientists at the library and military personnel at the bar (but they will always be there, even after solving the mystery), and while in StrangerVille (whether you're living there or just visiting), you get a new social interaction to "ask about StrangerVille".
From my experience, the infection can only autonomously befall NPCs in StrangerVille. I've never seen any infected running around in other towns, or my own sims getting infected while I wasn't playing them. A played sim can get infected by eating the strange fruit or breathing in spores in the basement of the laboratory; if there are other ways, I haven't discovered them yet. The infection itself is a moodlet that occasionally pops up, making your sim look and walk weird and keeping all their needs filled until it wears off.
If you solve the mystery, it will indeed be gone forever unless you restart it by... I don't know if I can say that, because it might be a spoiler for you. Let's just say, after solving the mystery once, you get full control over how strange you want StrangerVille to be. But as I said, it will definitely start out with a little strangeness, and there's nothing you can do about that (at least not without mods) other than going with it and playing the story.
The new military career is not tied to StrangerVille at all, though. Any sim in any world can join the career and get to the top of it without once setting foot in StrangerVille.
