Don't rush yourself in your enthusiasm; pushing too much can backfire.
Realize that horses are very adaptable. I had a woman student who had Parkinson's Disease. Of course she had never ridden before, so she didn't have high expectations. Still, she learned to walk, trot, turn, stop, and back. We had to modify the cues, but she worked out a language with the horse that satisfied them both. Some horses learn different "languages" better than others of coarse, just like some people.
You should feel no necessity to post. While posting can be beneficial at times, a good rider should learn to sit the trot. A sitting trot allows greater contact with the horse and, thus, greater access to communication. Much subtlety in riding can be achieved through slight changes in balance and pressure when sitting in the saddle.
If your horse is particularly bumpy, you might be able to lighten your seat a little by putting a little extra pressure on the stirrups without actually posting; you need not even show light between your seat and the saddle. It is important, however, to keep your knees relaxed. In fact, you should try to release any tension within the muscles around your pelvis and throughout your legs and feet. This includes the muscles in your crotch.
Much of the problem people have sitting the trot is caused by tension. Tense muscles are less able to absorb the bumping of the trot. Tense muscles in the crotch raise the rider's center of gravity and make the rider's seat less stable. When a rider releases tension in his muscles, gravity is allowed to do its job. The rider's seat softens and flows into the saddle. The rider's legs wrap naturally around the sides of the horse much like a cooked noodle would wrap around the sides of a bottle lying on the kitchen counter. There is no squeezing, only adhering. As a rider's weight is drawn downward by gravity, his center of gravity is lowered making his seat more stable. Gravity pulls downward on the rider's feet creating a firm contact with the stirrups. Generally, gravity would continue to pull the rider's heels lower than the balls of his feet. In your case, this might not be possible, but there is little to worry about. The flexibility of peoples' ankles, tendons, and muscles vary. The important thing is to relax and let gravity do its job.
Releasing tension in the muscles also allows the rider to better follow the motion of his horse's muscles as it moves. When a rider is without tension, well balanced, and moving with his horse, the horse can relax. His muscles will soften and his movement will become smoother and more flowing. This makes the horse's movement even easier for the rider to follow.
If you concentrate on these things, I think you will become excited about the improvement in your riding and forget about the limitations created by your surgeries.