Just a couple of small points from a cop in California.
First, when you "detain" someone as described above, there is no 20 minute time limit. It is not a concrete number, rather "a reasonable amount of time" to investigate the potential crime that you are detaining for (or to write a cite on a traffic stop for instance).
As far as jurisdiction, federal agencies generally enforce federal law, and have specific missions. So they wouldn't deal much with what you would call "street crime". Of course there are some exceptions, especially when you're dealing with inter-agency task forces (FBI-LAPD-LA Sheriff etc) In California, every peace officer has the exact same power (LAPD, Sheriff, Highway Patrol). It's just that they have primary responsibility for different areas.
For example, in Los Angeles -- where I work -- LAPD handles the actual city of Los Angeles proper, LA County Sheriff handles all unincorporated areas of the county, along with any cities that choose to contract with them instead of start their own police department, and California Highway Patrol handles the highways and all traffic duties in unincorporated areas of the county.
However, California peace officers have NO authority to make stops or enforce the law in Nevada or Texas etc. as stated above. The authority is state-wide, but doesn't extend beyond that to other states.
Hope that answers some questions and feel free to ask any more.