So there are 3 main types of drill you may be interested in. The first is a "regular" drill with a chuck (1/2 inch is common here in the US). The chuck is adjustable, so you can use a range of bit types. This will let you drill through wood and other soft materials, as well as drive screws through softer stuff. This is the most versatile drill - most novice DIYers can get by with this alone, although I wouldn't really recommend that.
The second is a hammer drill. These are good for drilling through concrete. They do have a bit of a rattle, but you won't have to use as much force on the drill, since the hammering movement of the drill does that for you.
The last is a impact driver. These also have a bit of a rattle, but are good for driving screws through thicker/harder material. This is like a hammer drill, but the hammering motion is in the direction of the screw threads.
I wouldn't worry about brushless - the main benefit is that the motor lasts longer, so they're good for professional use. I've never had an issue with non-brushless tools.
I'm not sure what country you're in, so I don't know what brands are available. Also not sure about how much you are looking to spend.
That being said, for simple DIY stuff around the house I'd pick a brand like Ryobi or Dewalt. Depending on what you want, you can probably find a combo set with a drill, impact driver, battery and charger for ~$100.
I use traps in combination with spraying their holes. Is that trap in your picture solid wood?
The traps kind of work for me. I made 5 and hung them up. Some are full of bees, and others are empty. Not sure if that's because of how I made them, or the location...
Also I've heard that if you put a dead bee in your trap, it'll attract others. Not sure how true that is though.