Do you acquire Evolution cards simultaneously when Attacking, and when can you use them?
In the absence of any published clarifications from the publisher or game designer, I think the rules for any game should be taken as literally as possible, unless doing so "breaks" the game and there is a more reasonable way to interpret them.

Naturally, should all players agree to a modification of the rules prior to the start of the game and it isn't some sort of "tournament" play, then the agreed on house rules will supersede the normal rules.  It is your gaming groups luxury to play how you wish. 

So let's take a look at the turn order from King of Tokyo rules:


Rolling and rerolling the dice
Resolving the dice
Buying cards (optional)
End of your turn.


From the King of Tokyo Power Up! rules (emphasis added):


  If a monster ends his rolls with at least 3{Heart}, he draws the top Evolution of his 
  deck


So this all hinges on when the rolls end. For me, rolls are clearly performed in stage 1, and the rolls would end before stage 2 starts.  As such, this is where the rules indicate you should draw the evolution card.

The following question was raised in the comments:


  "2. Resolving the Dice - The symbols you get at the end of your rolling step are the actions for this turn." You don't get the benefit of any die faces until you resolve the dice in the 2nd step, why would evolutions be different.


This is fallacious reasoning.  Nowhere in the rules does it say that "you don't get the benefit of any die faces until you resolve the dice in the 2nd step." It simply says that the symbols are your actions and the actions are resolved in stage 2.  A player doesn't take the rolls into the resolution stage, they take the symbols (and the actions they indicate) into the resolution stage.

Drawing the evolution card is not an "action" as defined by the rules nor is it coupled to performing actions. Evidence that the drawing of the evolution card is decoupled from actions can be found directly in the Power Up! rules:


  The hearts do not need to to be used to regain life, and a player can even use the hearts 
  to draw an Evolution when in Tokyo. The card does not replace healing, the card is in addition to any healing that would normally occur.


What I take from the above rule:


You can draw the card whether or not you can make use of the healing action.  Said another way, even when in Tokyo where the {Heart} symbols would otherwise provide no action.
You can draw the card whether or not you choose to make use of the healing action.
The drawing of the card does not replace the healing action.


In summary, drawing the evolution card has nothing at all to do with the healing action itself.

Moving on to if you can play the evolution card on the same turn, this is what the rules say:


  Monsters keep their evolutions secret until they want to play them – which can be at any time.


There is no mention of not being able to play the card the turn you draw it, so when they say anytime, you should be able to play it immediately if you choose.

Based on this reasoning, in respect to your specific questions, here is how they should be interpreted:


King of Tokyo: you can play anytime (for instance before you resolve the dice roll).  But to get the effect, you would need to play it before your opponent yields, but not after he/she does.  Once they yield Tokyo, you now control it so playing the card afterward won't provide the effect this turn. 
Alpha Male: if you play it and attack this turn, it will provide the effect this turn.
Chest Thumping: yes, you can use it to force your opponent to yield Tokyo.  You would need to use this when you attack (there is no point if your opponent has already yielded).
Jungle Tokyo: you could play it this turn if you did attack and didn't take control of Tokyo.


Edit: Addressing the point made by user1873 regarding Gigazaur's Gamma Breath/Saurian Adaptability.

Prior to this, I was representing (at the very least in my own head) the evolution draw as somewhat of an added stage in the turn (say 1.5).  I haven't sat down with the rules and all the cards to try to figure out every nuance, but it would have been more clear and accurate to say the draw take place in stage 1, following the rolls but before stage 1 ends.

The ability to modify the results of the final roll indicates the game creators have already allowed for game play to take place in stage 1 after the final roll but before stage 1 ends.  So I would take this as stage 1 doesn't end until stage 2 starts (i.e. the player moves on to start resolving his actions).

This allows for Gamma Breath to function (ignoring whether it is on the same turn played or in future turns).  If stage 1 were to end with the final physical dice roll, there we be no way to use this mutation within the rules as pointed out by user1873.

This also then allows for the evolution card draw as well as being able to play evolution cards at this time.