I have questions about Hard Forks.

Mainly trying to understand what happens to the number of available coins after a fork.

Say all 21m Bitcoins were minded. then a Hard fork happens, and for example half of the nodes go one way the other half go the other way. now what?

do the number of coins split in half? and now both chains now have 10.5m coins to mine? and do i as a coin owner have the right to choose which chain i stay on?

does the 21m bitcoins remain, and the new fork has new 21m to mine?

if these questions seem stupid, thats how little I understand of this topic. also i tried to search for an answer, but my question was too complicated for google, and i didnt really know what to ask.