Run or no run
 

OBR
Bases loaded, 2 outs.
Ground ball to shortstop, R3 advances to home, R1 moves to 2nd, and R2 has brain freeze and runs halfway to 3rd, turns around and goes back to 2nd.
Now there are two runners on 2nd base. Instead of just going to third and tagging the base, shortstop goes to 2nd and tags both runners. Out #3.
Is this still considered a force, regardless of who SS tagged first? Run doesn't count?

If both runners were touching the base when tagged only the tag of the forced runner would count no matter what order they were tagged. No run scores.

Yes.
Tagging R1 would not result in an out since he was on the base he was legally forced to advance to.

"force out" and "tagging the base" are NOT synonymous. You can get a force out by other than tagging the base, and you can tag the base to get an out when it does not result in a force out.

If the runner is forced from the base, then it's a force out, no matter how it's obtained. If the runner is not forced from the base, then it's not a force out, no matter how it's obtained.

No run. Because a tag was made on the runner being forced to advance from 2nd to 3rd prior to said runner advancing to and touching 3rd then the 3rd out was made on a forced runner and no runs score.

Great point, the one you see on this from time to time is when multiple runners are on base with runner at 3b, fly ball to the OF, runner at 3B legally tags up and runs home but some other runners leave early in the chaos of where to throw etc, the defense finally throws back to say 1B where the runner left from to record the out for leaving before catch, and even though you simply have to tag the bag in that scenario its not a force out, run counts.

I cannot count how many times I have been told that if a runner is thrown out at first base while trying to return after leaving early on a caught fly ball that it is a force out.
Thankfully this exact situation is spelled out in OBR under the force definition--where it clearly states that this is NOT A FORCE.

Its because its the one other out you see a lot on a runner thats not a tag play. Most people associate tag and force with how the out is made, not really using those terms to apply to the runner being "forced to run" or not. So they equate any out that is not a tag as a "force" but part of that is because its really not so much an action/live play although it happens that way as it is an appeal play.