How to play on Minecraft Servers on different versions

No, this is difficult to do. Every version of Minecraft will run slightly different, and some very different compared to other versions. If you want to play with the most correct mechanics in that specific server, you should use the optimal version that Hypixel (or any server) specifies, otherwise your gameplay might be effected negatively. In fact, some servers won't even allow you to join unless you use their recommended version of Minecraft.

As another note, it is a fully serverside choice, meaning you cannot do anything about a server not supporting a version.

You bought Minecraft Java Edition, period. You installed the version 1.17. But you have access to almost all versions, dating back to 1.0 and Beta. All you have to do is to open the Launcher, pick the 'Installations' tab, 'New installation' and pick whichever version you want from the drop-down menu. 'Create', return to the 'Play' tab and pick the new installation from the drop-down left from the 'play' button, and you'll play that version in a moment.

It takes a bit more work to have different installations differing with mods, if you install any, and there are serious issues with playing single player worlds on mismatching versions but that doesn't concern you now. You can play on Hypixel as the server authors intended.

In particular, Hypixel is unusual to even allow multiple versions (1.8 - 1.16.5) - most servers require you use installation that matches the version perfectly, and won't allow you to log in with a different one. So in short, you can't make the server support all versions, but there is absolutely no need for that - it's trivial to get an installation matching the server's version.