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I can't speak for all mic arms but mine (the elgato LP arm) came with two adapter screws.
Also,according to the article below, your mic should have already come with an adapter as well.
https://voheroes.com/how-to-attach-the-at-2020-to-a-standard-microphone-stand/
I think you're referring to shock mounts. These aren't generally one size fits all, you'd need to find one that supports your mic. The screw connection from the shock mount to the boom arm is pretty universal though (and if not universal, usually adaptable)
For the AT2020, you may want to consider something like the AT8458a.
As far as arm goes, and I know it's pricey, but the Rhode psa1+ is fantastic. Really smooth operation and a nice neoprene cover over the pinchy bits.
My only issue with it is that when it is in the stowed position it requires space behind the 'C' of the mounting clamp. Only an issue if you are installing on the back of a desk that is directly against a wall. I ended up moving mine to the front of the desk and no issues since.
Mic arms will have a tripod style screw mount at the end and then will probably come with several different pieces that can fit on the end. The one you're describing is a shock mount and they should list the microphone diameters they accept. You'll probably also be able to directly mount the microphone or use a rigid clip.
I bought this one a few years ago to use with my Samson Q2U: https://www.amazon.com/gp/product/B07X1ZWS2H The shock mount doesn't fit that microphone but the clip mount does.
Boom arm is the big arm the mic goes on. Mic shock mount is what you're talking about, this reduces noise from your desk. Hiss / pop filter is the fuzzy thing on the end
