Can I delete Employees from a cloned budget without affecting the original?

The short answer is yes you can.

When you clone a budget, you create a new version that is completely separate from the original. There are no links, so any changes you make in the clone will not affect the budget you cloned from.

The Employee Module itself is unique to the budget in which it lives, so you can have a completely different group of employees and HR formulas in each budget if necessary.