Rules for algebraically manipulating pi-notation?

You just need to think about what the product notation means, and you can work out the rules yourself.

Take one of your examples:

$$\prod_{i=1}^I x_i y_i = \left(\prod_{i=1}^I x_i\right)\left(\prod_{i=1}^I y_i\right)$$

This rule works because the left side multiplies $x_1$ times $y_1$ times $x_2$ times $y_2$ etc., and the right side multiplies the x's first, then multiplies the result by the y's. Clearly those are equal.

Another one of your examples:

$$\prod_{i=1}^I \exp x_i = \exp \left( \sum_{i=1}^I x_i \right)$$

Can you figure out why this rule works?