Can Feds tamper with evidence in State police lockups? How much information are they cleared for with regard to satelite information? How about foreign travel to destroy a world-wide cult?

Can Feds tamper with evidence in State police lockups?

Not legally. Neither can the local police, for that matter, though it certainly does happen on (hopefully rare) occasion. If the evidence pertains to a federal investigation, federal agents may have access to it for the purpose of conducting tests or examinations or whatever, but the scope of such activity will be circumscribed so as to protect the evidence for eventual use at trial. The importance of maintaining the "chain of custody" is drilled into police investigators at every level, though certain representatives of the LAPD seemed to forget the rules, with very public consequences, a few years back. (Hint: if you find a bloodily murdered corpse and your techs have not yet inspected it, DON'T COVER IT WITH A BLANKET!)

Criminal evidence lockers tend to be fairly isolated in police stations, but not always. I was at the NYPD evidence labs for a tour a few years ago, and the labs were overfilled from a recent massive bust so there were shrink-wrapped blocks of marijuana piled four feet high in the hallway. Only police personnel could get access to the halls, of course, but federal agents could easily get in as long as they say they are pursuing some legitimate investigation, and it's conceivable that important evidence could be easily accessible once you are inside the building proper and as long as you are wearing a name tag that says "I belong here".

Note, by the way, that in the U.S. federal agents can NOT boss state or local agents around except under very specific circumstances; consider them parallel organizations, each with its own jurisdiction and authority. Of course, when an investigation involves multiple agencies (kidnapping, for instance), a federal agency may be designated the "lead" agency for that investigation, in which case its officers may direct the activities of the state and local police. But that's not an automatic thing, and the state and local police sometimes don't particularly care to be directed by the feds.

How much information are they cleared for with regard to satelite information etc?

Federal law enforcement officers will probably not have easy access to satellite imagery, though I could conceive of a well-connected official placing calls to get weather or commercial satellite images; military or intelligence satellite imagery, under the purview of the NRO, may be much more impossible to obtain. As described in DG, federal law enforcement officers have a totally different set of "Secret/Top Secret" clearances and regulations than intelligence or military personnel.

How about foreign travel to destroy a world-wide cult?

Some federal agencies conduct foreign travel all the time: the CIA and the State Department, obviously, and some in the Departments of Justice and Treasury. The DEA and the Customs Service are most likely to be traveling abroad. The FBI traditionally has traveled infrequently (WW2-era operations in Latin America notwithstanding), because their mandate allows only domestic operations. An FBI "legat" (legal attache) can be found in your larger US embassies. But in recent years FBI agents have gone abroad much more often as advisors, particularly to the former Soviet Union. The U.S. Marshals service are the official Interpol liaison for the United States, and they are charged with transporting international fugitives back to the US; recently the Marshals brought one of the World Trade Center bombers back after CIA and FBI and local agents captured him in Egypt. But as you can see, federal law enforcement agencies have tight restrictions on their international operations. Delta Green agents from the FBI or other law enforcement agencies would be better served to obtain a military or CIA or State Department cover.

And will Delta Green give them experimental weapons, or bail them out of jail?

This is definitely a "Case Officers preference" question. In some Ops, DG will not have much in the way of experimental weapons, unless the agents steal something Grey-related from NROD agents.

Getting bailed out of jail: I would say that if DG can do it through a legitimate cover, then cool. But if doing so would elicit questions likely to lead to a Delta Green cell, then the imprisoned players may be SOL. "Do not jeapordize compartmental security."