1a) I understand the cash is instead of .57 RIG shares (.57 x 100 = 57 RIG shares) and is fixed (does not vary with the price of the underlying shares of RIG).
I'm not sure that you do understand it. They don't issue fractional shares in these mergers. So the cash in lieu amount is approximately $74 (.57 shares of RIG times the settlement value of approx $129 )
1b) But if a person wanted to exercise the 80 call, does the strike price for the 80 call remain at 80 for the 47 shares of RIG (the shares that do fluctuate that the price of the option is based upon) that a contract owner would receive if he exercised the option?
The strike and multiplier stay the same. A Jan 08 80 call option gives you the right to buy 47 RIG shares and receive $2,246 and receive the cash in lieu amount of $74 for your $8,000. Your cost basis for the 47 shares is $8,000 - $2,246 - $74 + premium paid for the call.
2.) Is exercising the call the only way to receive the cash portion? I.e., if you simply sell the call, no cash is received except for the price at which you sold the call for?
What's the point of trying to get the cash? You're giving them an extra $2,246 to get $2,246 back.
Think of it this way (a wacky make believe example). You own a call with the right to buy 100 shares of XYZ at $90. The merger shareholder sells 40 shares @$90 ea and gets $3,600. If you exercise, he gives you 60 shares plus $3,600. In terms of value, 100 shares at $90 is the same as 60 shares at $90 plus $3,600. Obviously, there's more to this in the real world but the cash you have your eyes own is your own money
3.) Why cash instead of .57 shares. Shouldn't it be .53 shares, so when added to .47 shares of actual RIG shares, the total becomes 1 share (since the multiplier remained @ 100) ?
As mentioned above, it's a fractional share. The terms of the merger provide you with 47.57 shares. Since they do not issue fractional shares, they give you 47 shares and cash in lieu of the fractional amount (.57 times my guesstimate of $129+ per share).
Clear as mud?
(I'll be amazed if I didn't typo some numbers wrong in this)