Failure to drive those displays, big TVs or projectors for the most part
I thought they could drive up to 6k or something crazy - surprised to hear it struggles with TVs and projectors that I imagine are 4k.
They don't have HDMI port, so you're limited to your adapter or the TV's display port capabilities.
If you're dependent on the adapter, you need to use the Apple ones, or you need to have multiple. Sometimes HDMI works, sometimes it doesn't, changing the adapter usually fixes it.
Sure, but there are plenty of adapters that drive 4k@60Hz without an issue, such as the latest revision Apple's adapter (old revisions would only do 4k@30).
What do you use to connect the screen to your Air? I've got an M2 air, and recently purchased a 5120x1440 screen with HDMI and DisplayPort outputs. I first connected it through HDMI to one of the USB/HDMI/USB-C Apple adapters, and the result was terrible. But I ordered a DisplayPort->USB-C cable, and it worked perfectly.