Honestly if it needs an app it's probably overpriced garbage anyway. High quality analog headphones have been made for decades without any apps. As others have said ditch the "gaming" headphones, they almost always suck and are overpriced for what they offer.

You should ideally look for headphones with a frequency range of 20Hz to 40kHz (the full range of human hearing) but even a 40Hz to 20kHz range would be decent. This is the biggest quality factor for headphones.

If you're gonna plug it straight into your computer or audio card via aux or usb you want something with low ohms (either 2Ω or 6Ω) but if you're gonna use an audio interface or mixing board you want something with around 200 ohms. In laymans terms, low ohm headphones with powerful audio device = blowing out the headphones. High ohm headphones with tiny device (like a phone) = quiet muffled audio.

For microphones, range doesnt matter as much because most people talk around 200Hz and don't really go too high or low. Also look for "condenser" mics which capture voice much better than dynamic mics.

For connection, best quality is probably gonna be an audio interface. The mic connects via XLR cable, the headphones over aux, and it usually has a USB-A connection to the pc. If you're running Linux, Pulseaudio has great support and is much better than default ALSA.

My favorite mic brands are Shure and Rhode, and my favorite headphones are made by a German company called Beyerdynamic. Mackie also makes some cheap and quality mixing boards. Probably more info than you needed but the audio rabbit hole is very fun to explore!