why did the Don and Michael have Rocco put together a secret regime? what was the regime supposed to be used for?

It was kept secret so the other families continued to believe the Corleones were weaker than they really were. To pull off the killings of the heads of the five families the way they did, Michael needed to be a lot stronger than he was when he returned from Sicily. Also, once it became clear that either Tessio or Clemenza would betray Michael the Lampone regime was absolutely critical.

I don't believe it was "clear that either Tessio or Clemenza would betray Michael". Lampone with his powerful regime would know that he held a key position. If Paulie could betray Vito, why couldn't Rocco betray Michael? And if you're right about the covering of Michael's strength, why not letting Tessio build up a secret regime? If Tessio had known what was going on, he probably wouldn't have betrayed Michael.

But that's the point D. Tessio had to have a reason not to betray the Corleones. That's certainly not loyalty. Tessio and Clemenza were Vito's people. Despite the fact that Lampone came out of Clemenza's regime, it was Mike who gave him his command. That bound him to Mike and minimized his betrayal potential. In the end, Mike had to make a choice about whom to trust.

I think that line about Rocco building a secret regime was careless writing. Rocco was Clemenza's man, not Michael's. And it would be difficult for Rocco to build a secret regime under Clemenza's nose without Clem knowing about it. Since at that time Michael didn't know who would betray him, it would be a signal to Clem that he didn't trust him.

As I posted above TB, it was Mike who gave Rocco his command, not Clemenza. Thus, his loyalty would be to Mike. Also, the building of the regime was a "secret."

When you think about it what Michael was doing in addition to making the family stronger covertly, was bringing "his people" into positions of power in anticipation of Vito's imminent death. That's generally what happens when any new CEO takes over a company. Clem and Tess were "Vito's men" and their loyalty was to him, not Michael. With the rise of Rocco and Neri, what was happening was Rocco got a promotion by Michael, and Neri was 100% Michael's man. That's why those were the two who went with Michael to Tahoe, and why Michael left Clem (succeded by Frankie) back in NY to run the olove oil business.

Clemenza was not as smart as Tessio, but you would think he would have had to known something was up. Rocco would have gone from being Clemenza's new factotum to not being available to Clemenza very much at all.

That Clemenza and Tessio are both worried that Michael won't give them permission to recruit new men show that they were probably unaware of Rocco's new crew.

Rocco's new people would have had to make a living somehow. In the book they are "on payroll" but Tom thinks they make too much money for their alleged jobs and figures out the deal. In the movie would they have been responsible for their own livelihood? How were Rocco and Mike binding these new guys to them?

You answered your own question: they were salaried. It is not clear if they were permitted to engage in any other money making activities that could expose their association with Rocco, however.

Very difficult indeed
Besides how would Rocco explain his absence / having little to no time for any jobs Clemenza would have wanted Rocco to do....

couple more threads [could be others] which maybe of interest
Secret Regimes
Rocco's secret regime

careless writing to make Tom look smart finding out the secret regime

Wouldn't Vito's undertaking that he will not be the one to break the peace pact, binding on all the Corleone family

No. Michael wasn't at the Commission meeting, when Vito swore that ..."I will not be the first to break the peace we have made here today." Vito died without breaking the peace. Afterward, Michael figured, correctly, that Barzini was going to break the peace by killing him.

This maybe of interest
Breaking the peace?

Barzini presented himself as the mediator, but the peace was actually between the Corleone and Tattaglia families.