a "fully" American option strategy?

A "fully" or "just" American strategy would include calculating for possible option assignments. In which case your possible loss equals option's intrinsic value (plus commissions), which is less than the option's market price, so you can immediately re-sell the same option, capture the difference and restore your position. You lose on commissions, spread and slippage and likes but gain on time value of the newly sold option. Given that simple trick, there seems to be little difference.

Yes American versions of options spreads are traded very commonly. Some brokers such as ThinkorSwim show you american equity options spreads that are traded as one order and you can see they are very common.

Being exercised on your short legs is mostly advantageous to you, because you get all of the time value, early, and will still be hedged by all the shares/underlyings that you get.

Although it messes up your margin requirements, you don't have to adjust those instantly (you usually get 3 - 5 business days), and you are still hedged, so you should be able to get back into the position if you still want it because your short options will be covered by your long options still. You'll lose on slippage and commissions.

A butterfly spread contains a bull spread and a bear spread. It can be comprised of 4 calls or 4 puts or 2 calls and two puts.

So while you got the part right about 2 calls and two puts, the rest of it is all wrong since you have not created any spreads. IOW, you're long two puts and naked two calls.

Instead of X, X-a and X+a, let's pick a strike for "X" (say $100) and pick an increment for "a" (say $2). So now you have:

• long one 98 put
• short two 100 calls
• long one 102 put

If you combine a call with each put you get:

• +98p - 100c
• +102p - 100c

Each of these is a non standard synthetic short position. So you have effectively shorted 200 shares which functions normally below $98 and above $102 (it makes 2 points for every point XYZ drops below $98 and loses 2 points for every point XYZ rises above $102. Between $98 and $102 it's a little different but if you understand options, you can do the math.