I am about to try outdoor trap that has very strong odor.
One of those
https://www.amazon.com/Victor-M380-Magn ... s=fly+trap

Do you think it would affect any wild birds? The smell is disgusting, everyone says.

The wild birds probably won't come close enough to it to be affected by it.

hmm.... it depends where I will hang it. What if it's on the tree, and they already nest there? A problem is that it's not easy to choose a location for it. It says not to put it close to the house... but where it's not close to my house, there is somebody's else house

Well don't put it right next to a nest, or any other place where birds tend to congregate. Pick out a spot that's likely to attract flies but not birds or other animals.

I have a similar fly trap up. It draws lots and lots of flies in. giving you the idea it works great. But I'm going to tell my wife not to buy them again.
I believe the reason you can't put it close to your house is because it is so good at luring the flies to the trap some will get into the house instead of the trap.
And it does smell...

thanks Bob. Why will you tell your wife not to buy it again? The reason to keep it away from the house, I believe, is the smell.
Need your opinion... if I have rabbits outside and want to deter fly from where they are... should I hang trap near the bunnies or away from them? It says place it near their common areas such a garbage, etc, means that the idea is if they are trying to get into the garbage, they would fly into the trap instead... following this logic, I need to have it near the bunnies LOL

Do the instructions say anything about it being hazardous to animals that smell it? Are rabbits very sensitive to smell? If the answer to both questions is "no" then you can probably hang it near the rabbits. But it's best to be cautious with it, and gradually move it closer to the area where you want it so you can watch for possible problems.

I don't want my wife putting it up near the house again because it draws in more flies than when it wasn't there.

As for where to hang it, the smell is weird, but you get (or at least I did) used to it. I'd follow Caro's advice, move it gradually closer untill you find the sweet spot...

the more I read reviews, the less I plan to get it.
No, it works fine for more than 60-70% of the people (which is HUGE success rate IMHO)
But those who are in minority, say horrible things about it. Like basically instead of getting rid of flies, it draws more and becomes a breeding site for them (as maggots can escape the trap)