Adam.. Looks like you have plenty of room to work, the lack of mechanicals will make any modifications far eaiser. I'll echo some of the suggestions already made for improving the performance of your floor system, and throw in a couple of other options... Just depends on what lengths you want to go to..
Many of the same principals for soundproofing also apply to improvements in your floor and the "Bounce" you feel.. like mass, dampening and stiffness.
Ill throw this out quick before talking about your joists, just for something to keep in mind for the future..If you ever decide to change the flooring above this room, and you can stand and additional 3/4 inch, I would throw down an additional layer of T&G perpendicular to the original decking with a layer of GG in between.. this will help tremendously in further reducing "Bounce" and any impact noises in or sound out of the theater.
Although deflection and bounce are 2 different things, stiffness of the members impact both. So the goal is to increase the stiffness and deflection is a means of measuring this.. If I was looking at this, I would check to see the amount of deflection the joists have settled into under the current static load.. Eaisly done by pulling a string along the bottom side of the joists and measuring the deflection or "Sag" at midspan. the minimum IRC codes for living areas is L/360 and this would allow ~ .75 max, but in the real world this could still be a very bouncy floor and meet all codes. So assume we want a stiffer floor and go with an L/480 for deflection. This means we would want to keep the the "sag at midspan" under .5375 or basically a 1/2".
If the measurements were more than this 1/2" (particullarly on the spans down the middle of the room). Then I would seriously consider Supporting or "Jacking" up the joists close to midspan before doing any retrofitting..
Im not advocating a permanant post and beam(although a definate fix to the bounce issue) becasue I understand you are already concerned with height, but this is temporary while you make the improvements to your floor.
Note: depending on the amount of deflection your seeing and how much the joists have taken a "Set" this may need to be done over a period of time.. ie jack them a little at a time and let them set, then jack a little more.. let them set.. etc.. etc.. how much you move them at a time and how long they set will again be a function of how severe they are sagging now.. you'll need to make that call as you go.. you can continue to use the string along the bottom cord to see when your where you want to be. (I would reevaluate this if i had a tile floor above)
Now assuming we removed most or all of any sag that is present.. the first thing i see is (or dont see) in your pictures is that when they put the double joists in they didnt nail them in any manner that would be considered "Sistering" by normal standards.. this is typically done by nailing either 2 or 3 rows of 12-16D nails on 6-8" centers (typically 3 rows for 2x10 or wider) for the length of the joist. Even 2 rows on 12" centers would be an improvement .. If this is already done, and I just dont see them in the photos then disregard.
this lack of fastners probably contributes a signifigant amount to the squeaking you hear as the joists flex and move against each other..
(Ideally you would have also put construction adhesive between the joists)
As long as I was going through and adding the nail pattern to the joists, I would rip down some 3/4" Plywood just shy of the current joist depth and glue and nail it to the sides of the joist as IN2 mentioned. The minimal cost of the ply makes this an attractive retrofit.. you dont need to do this the full lenght of the joist but I would consider the middle 12ft as a min.. (alternating a 4' and 8' pc) you also dont need to do every joist, but the more the merrier.. every thing you do will add mass help stiffen your floor and reduce bounce.
Next I would use solid block Bridging at a minimun of the 1/3 span points, I would cut these from 2x10's with the goal of getting them to fit as tightly as possible within reason Try to avoid lumber with much moisture because you dont want shrinkage later to cause squeaks, it will be harder to nail but keeping the blocking in a straight line is better than staggering them, you can use construction adhesive as Milt said to help keep future squeaks to a min. This will help stiffen by transfering some of the loads to neighboring joists.
Another technique to stiffen the floor is to use metal strapping on the joists.
if you do this you want to do it before the blocking/bridging and while your floor is supported with the temporary jacks.. I looked today but couldnt find a picture of what I'm describing.. Using strapping like the simpson strong tie CS18 or CS20 you basically start the strap on an "Upper" end of the joist.. this strap then wraps down around the joist crossing the bottom at mid span and back up to the Upper opposite side of the joist at the other end.. critical factors when applying the strapping would be the size/type of fastners and making sure to keep the strapping tight aganist the joist as you go. What this does is help transfer some of the tension loading from the bottom and middle to the top and ends of the joists..
Sorry about the ramble
... you have alot of options to choose from.. some or many may consider it over kill, but probably what I would do
hope this helps.