Learn Swedish: Jag har inte satt på skivspelaren på en evighet. - I haven't turned on the record player in an eternity.

Because the sentence is negative (inte). When you state how long something hasn't happened, Swedish uses på for the duration. If this were a positive sentence about an ongoing action, like "I played records for an eternity", you would use i (i en evighet).

There are two pås in this sentence. Are they doing the same thing?

No, they have completely different jobs! The first på is part of the phrasal verb sätta på, which means "to turn on" (a device). The second på is the time preposition working with en evighet to show how much time has passed.

Is satt related to sitta (to sit) or sätta (to put/place)? I always mix them up!

Here, it belongs to sätta. The phrasal verb is sätta på (to turn on). Sätta conjugates as sätter, satte, har satt. It is very easy to confuse them because the simple past tense of sitta is also satt (sitter, satt, har suttit), but since we have har here, we know it's the supine form of sätta.

Can I move the first to after the object, like "Jag har inte satt skivspelaren på"?