I read somewhere that if you use running shot, your Ranged Combat Expert skill has no affect with you shot. First, is this true? Secondly, Charge is very similar to running shot, so if you use it, do you lose any powers like super strength, or blades/claws/fangs?

You can't use RCE or CCE with any other super powers. Only team abilities. Charge can be used with blades and super strength. The PAC states:

RANGED COMBAT EXPERT. This character adds 2 to their damage for successful single-target ranged combat attacks. This power does not work with other super powers (such as Super Strength or Enhancement).

You are correct with Running Shot. You can't use RCE at the same time you use Running Shot. Similarly you cannot use CCE at the same time you use Charge. You can use Super Strength and BCF with Charge, however, as those powers do not say they're not usable with other powers.

The simple rule is no other power my be used by or on the character that wants to use RCE/CCE. The only exception is Probabilty Control. Another character with Probability Control may allow a character using RCE/CCE to reroll their attack.