[cosmicfish]

This looks like one of those areas where you need to be careful not to ascribe prescience to the planners - remember that the planned deployment of the Project would occur in a time when there were going to be scattered remnants of the US military as well as a lot of veterans among the survivors. Much of the work and materials and expertise would already be available and the US would not generally have descended into mutual hostility. I think that restoring the military would be a high-level function, not one parsed to individual teams beyond a very basic, temporary level.

That having been said, some comments:

What is needed?

Considering that the Project is presumably planning on (a) dealing with immediate threats itself, (b) working to restore the United States Military under some new national government, and (c) notoriously worried about arming people against itself, I would say the Project would encourage only a small defensive force and only if it was unusually necessary.

Who will make decisions about the Security Force?
What are the costs and how will they be paid?

Presumably, the goal would be to leave any such force under the control of local civilian government, but leverage the influence and resources of the Project to steer it in an acceptable direction.

As to costs, I think it likely that the Project would be encouraging a commune-style economy at the local level, until large-scale security had been established and actual commerce made capitalism feasible again. So at first... no one. Because no one is getting paid, everyone is pitching in to survive. For that matter, it is unlikely that any but the largest communities will have any full-time soldiers, not when basic survival takes so much work.

Is the existing force corrupt?

If it is then you have a bigger problem to resolve before you can even start improving their forces!

How do you get the best recruits?

It is likely to be a "whoever can and will fight" scenario. Beggars can't be choosers.

Does everyone understand that Security is more than "killing bad guys"?

For a solid decade post-war, anything more than "killing bad guys" is going to be escalated either to Morrow units or to reconstituted units of the US military. For newbies, stick to the basics.

Does this Security Force frighten the neighbours?

Well, if the neighbors are hostile then it may be necessary regardless, and if they are not then the team should have engaged them long before starting any scale of military development that they would know about or even care about!