Need a cassette here for an alexrim. How do I choose one? Need lockring too, will also need to know which one and do I need to consider what my derailleur is with the new cogs (for compatibility)[?
The depth of the cassette is related to the number of gears, but 8/9/10 all use the same width of cassette, simply making the chain thinner and space between cogs smaller.
A new cassette will come with a lockring, so that's covered.
You need to match the number of cogs with your shifter, and have a chain the "correct" thickness. There may only be half a millimetre difference but that's enough to make a transmission work or not.