For under $50, my guess would be everything will sound about the same in the $20 - $50 price range. Just search Amazon for something you think looks good and has a lot of positive reviews.
The biggest differences in that price range will be in quality and comfort of the ear pad cushions and the aesthetic. You will probably find a lot of great gaming headsets but they may have large ear cups and a flashy graphics -- like my teenagers have for playing games.
A lot of people don't care, some people and/or their co-workers may not like those in a professional environment.
The single vs double earpiece is a personal preference. You may have to go somewhere and try them out.
As far as the ear buds randomly connecting, You can use your phone, you might want to remove them from your list of paired devices in-between uses, or at least during your weekly meeting. Your phone is just looking for paired device periodically and if it sees the headphones available, they will just connect.
If the earbuds you own have active noise canceling, look through the manual and see if there is a mode that also lets you hear ambient noise. Many manufacturers offer it; some may refer to it as an ambient or pass-through mode. It allows outside noise in so that if someone talks to you can actually hold a conversation with them without removing your earbuds or headphones. You can also hear someone walking up on you or things like cars approaching.
