Replace the Spider: Doppelganger can use Exploit Weakness, but only to target characters named "Spider-Man" or characters possessing the Spider-Man team ability. When he does, the target cannot use team abilities and the attack cannot be evaded.

Twisted Clone: Doppelganger can use Combat Reflexes and Leap/Climb.

Minion: Mindless Pawn: Once per turn, if Doppelganger has exactly one action token, you may give a power action to a friendly character with a higher point value within 8 squares. You may now give Doppelganger one action as a free action.

If Doppelganger is attacking a character using Replace the Spider's Exploit Weakness, can that character use Shape Change or Mastermind? Basically, what exactly constitutes evading an attack?
Super Senses evades an attack. Some other special powers may also allow a character to evade an attack. Shape Change and Mastermind, though, do not use the word "evade" anywhere in their description. Shape Change prevents the character from being attacked in the first place and Mastermind transfers the damage dealt to another character.

So unless the words "evade" or "evades" appears in the game effect description, Replace The Spider will not have any impact in that specific regard (you could still use Exploit Weakness, though).

So I want to use Mindless Pawn. Can I give my more expensive character a power action to use Charge (or some other power) and activate Mindless Pawn?
No. You can not give a character more than one non-free action per turn and in this case you are talking about two separate power actions: one for activating Charge and another for activating Mindless Pawn. One power action can't be used to activate both.

But Mindless Pawn says I may give a power action to a friendly character with a higher point value. Didn't I do that when I used Charge?
Well, you did give them a power action to use Charge, correct. But take a look at the description for Charge:

Give this character a power action; halve its speed value for the action. Move this character up to its replacement speed...

Notice that Charge also states that you "give this character a power action". When you want to activate most any specific power or ability, you generally have to give a character an action specifically to use that power or ability.

If Mindless Pawn was meant to work as you ask, if it would indeed activate with any old power action given to do something else, Mindless Pawn would instead read more like "when (or whenever) you give a power action to a friendly character with a higher point value". It does not read like that, though, so you must give the higher point character a power action to specifically use (and only use) Mindless Pawn.