If I lower its stats below the requirements for the position, will my pet lose its job? And how low can your stats go?

I've never heard of a pet losing its job. As far as I know, you just need the stats for the promotion but it doesn't keep checking that those stats are still valid (that would be a waste of site resources).

As for how low they can go, you can take strength, defence, health, and speed down to 0.

Thank you so much! I am going to work on some pets now.