I've looked around but can't get a clear answer and hoped you could
help me. I can only access my hmc via ssh right now (I'm in a
class...). I need to know how many CPUs my managed system has and how
many are allocated to partitions and how many are available for cuod
operations... Can you help me?

I did: lshsc -i -a
to get the managed system name

then I did
lscod -m 9117-570*XXXXXX -t cap -r proc -c cuod
12

I think this means my system has a total of 12 cpus... is that right?

How can I tell how many are allocated to my partitions (to see if there
is any excess capacity available)?

I tried:

lssyscfg -r lpar -m 9117-570*XXXXXX
lssyscfg -r sys -m 9117-570*XXXXXX

Any help would be appreciated.