Whew, kinda of a short question, but I can only think of a long answer.
The commands should be run on the system that is running the database. So however you achieve that, is up to you.
To bad it's windows, or else you could simply make a distributed server setup on it, like I did with Solaris.
The check_oracle plugin is quite useful, and I check everything that it is able to, and graph the free table space in all the tables, which has been a lifesaver.
The Oracle_home thing is part of the environent. On Solaris, if I login as some user that is the Oracle admin, then ORACLE_HOME is set and shows the path to the oracle home directory. But you can't do that, on your local machine, since you don't even have a oracle installation.