Any tips on what client to use?
I like and liked Hex-Chat (once: X-Chat) it is a GUI-Client for almost every desktop system.
Is there one that works well on both Windows and Android?
I never used IRC on mobile productive. IRC works like: You join a channel, you ask your question. Than you wait typically some minutes, sometimes some hours and very rare you get an answer after a day or more. And when you got your answer/discussion/whatever you typically stay for a while, maybe to answer someone elses question or to learn from the discussions or to get a feeling how that channel works.
This is why IRC isn't so cool on mobile, you would want to stay in the channel like forever and you will quit when your phone gets to standby, so you better join with your desktop computer and just stay there as long your computer is running. There is an /away
function that shows others you are not at your computer.
For a similar mobile experience you'd need a more advanced setup with a "bouncer" (an IRC-Client that runs on a server and constantly stays in the chatrooms for you and when you connect to your bouncer it sends you what happened when you weren't there).
I think weechat runs on Windows and Android but it's pretty complicated to use (but very powerful to script).
To a lesser degree, what’s the relationship between IRC and #Riot / #Matrix, if any?
As far as I know there are bridges/bots that can connect Matrix and IRC Channels to each other.