Removal of pitcher
 

In a tournament this past weekend, the opposing coach made a mistake and made a trip to the mound which resulted in them having to change pitchers, they swapped the P and the 3B, who had pitched earlier in the game, which no issues there as softball (that I know of unless local rules) there is no limit to bringing a P back.

However, obviously he did not want to do this as the 3B threw one pitch, he went out again and swapped them back. I was always under the impression in a forced sub situation (like P throws warm up pitches or is on lineup card etc) that they had to face/complete one batter.

Once a pitcher is removed from the circle because of too many conferences, that pitcher can no longer pitch for the duration of the game. So your coach was not allowed to put the pitcher at third for one pitch, and then bring her back to pitch again, regardless if the other pitcher delivered one pitch or pitched to multiple batters.

As for a new pitcher's requirements, there is no rule that I'm aware of that says she has to complete one batter before she can be removed.

You didn't tell us what ruleset; so we'll give you incomplete answers that may not apply.

A couple of things to consider.

First, there is no mound on the softball field. There is a circle, but defensive conferences don't relate to the circle, either.
Second, if a coach must remove a pitcher due to an excess conference, that pitcher may no longer pitch for the rest of that game. Period. She's done pitching.
Third, there is no minimum requiring any LEGAL pitcher to maintain that position, except for the NFHS requirement in the bottom of the first if she utilized a courtesy runner in the top of the first. But you didn't tell us which rules, right?
Fourth, there is no rule requiring a forced sub if someone takes warm-up pitches, or even if listed on the lineup as pitcher (except as noted above).

Was in a USSSA tourney

Thanks, 20+ years on the baseball side, some of the rules start to blend together.

Thanks, interesting note with regards to courtesy runner, is that only if she is the Pitcher that inning, or if you plan to change pitchers the next inning on defense and your new P or new C for that matter gets on base, can you sub them out. On the baseball side in tournaments my son plays in, they have it backwards, the intent of the rule is a speed up rule, but you cant run for the P/C the next inning unless they were pitching the previous inning and you can run for both and not have them be the P or C the next inning, which to me is way backwards.

It's not apparent whether the coach realized it was his 4th trip out and made the decision to remove the pitcher himself or did not remove and the umpire directed the removal. If the coach made the decision himself, wouldn't he be allowed to bring back the pitcher. If the coach removed the pitcher it would not be a charged conference.

Under USSSA rules you are permitted three charged defensive conferences in a regular inning game. On the fourth conference a penalty is imposed. The penalty is removal of the current pitcher for the remainder of the game. If the coach announced the pitching change prior to going to the circle, he would have been permitted to change his pitcher and then bring her back later. In the OP it sounds as if the umpire imposed the penalty, but then erred by allowing the pitcher to return.

He didnt want to remove her, which is why he swapped her out with another P, who threw one pitch and then put her back in, which I did not think was legal, but umpires let it go, said there was no such rule prohibiting him from doing that.

On another note (same game) we were tied, bottom of what could potentially be the last inning if we score because time had expired, we had 2nd and 3rd with either 1 out or nobody out, and they wanted to intentionally walk batter to set up force at home, umpire allowed the verbal pass vs throwing pitches, but again different associations, etc Im not 100% on that one. Again it was USSSA

The umpire was wrong to permit the pitcher that was removed to return.

Under USSSA rules a pitcher must throw all four pitches for the intentional walk, they are not permitted to issue a verbal pass. With that said, sometimes local tournaments issue tournament specific rules, and may have allowed verbal intentional walks in the interest of trying to speed up games.