EDIT: I can answer all of this in detail but I'm off to work myself right now. keep this question going until this evening. The contractor's mistake was in how he did the sub-flooring, by the way, and if he's a bonded, legal contractor he's got to make it right.
EDIT: Sorry I got back to you so late, but I was actually out finishing a job. Also, John DMB pretty much gave you the right answers.
But I'll add this. Your house is on a pier and beam foundation. Chances are the walls are plaster over hasp over cedar studs. What your contractor SHOULD have done was remove the whole bathroom floor, including the sub-floor, down to the exposed joists. If there was no serious damage to the original joists then a new floor can be installed. However, if there was dry-rot or termite damage, etc. the joists should have either been replaced or at least new ones butted up against the existing joists and the piers either replaced, reposistioned or added to. And in either case the joists should have been plummed (made level).
Now about the floor itself:
The toilet was sinking because the boards under it were rotting out. Replacing a few floor boards would not have solved the problem, would not have gotten rig of the dry rot. The whole bathroom floor should have been replaced.
You floor would NOT squeak it he have used 3/4" plywood as a sub-floor and screwed it down with screws onto the joists. And then screwed down the cement board over the plywood sub-floor.
You have the gap for one of two reasons. First, building materials used back then were measured at true measure, meaning a 2x4x8 stud was actually measured at 2" by 4" by 8'. Now a 2x4x8 is actually 1 ½” x 3 ½” by 96 ½ “. A double layer original floor would have been 2" thick. Now that same double layer floor is 1 1/2" thick. Which explaines one reason for the gap. A skilled contractor knows this and makes the depth adjustments from shimmed joists or by installing both a ‘true’ measurement sub-floor and floor so that the new floor stays 'true' to the original floor.
Or the floor could have actually pulled away from the wall frames when the weight was added to the bathroom. If this happened it was also your contractor’s fault. Sampson stud ties are sold and used for just this reason. To anchor and secure the floor to the wall frames. This should have been done before the cement board and tile floor went down. You strap wall studs to joists at the corners and, if it's a larger room, in the middle, too. You do this before you even put down the new sub-floor. You would not have had a wall/floor separation if they had been anchored to each other properly no matter how much weight the new tub weights when full of water. Healthy and plummed joists, proper floor installation and wall to floor anchorage and you wouldn't have had the problems you have now.
EXTRA EDIT: I suggest you copy and print out John's and perhaps my replies to use as referrences when talking to your contractor again.