Which option is correct? and why?

"Drop her off" or "drop off her"
Pick him up or pick up him
find out something or find something out

"drop", "pick" and "find" are transitive verbs.  These verbs must have an object.

Intrasitive verbs are those that do not need a direct object after it.  For example, "to die", "to age", "to sleep"

Then there are "ambitransitive" verbs.  This means that the verb can be transitive and intransitive.  For example, "to play".

His son plays.  (intransitive)

His son plays the piano (transitive)

"Drop her off" is correct
Pick him up is correct.

"Up" can have various meanings depending on the context of verb. Usually adding the word "up" changes the meaning slightly. However, there are some cases where difference is not that great or even optional.

Find something out is correct.

"Out" has similiar usage as well. Here's a useful video that explains how to use "Out" as phrasal verbs: http://www.youtube.com/watch?v=WsE4UmmkVtU

This website has a lot of great examples: http://www.phrasalverbdemon.com/particleup.htm

Hopefully this helps.

"Drop her off" is correct
Pick him up is correct.

"Up" can have various meanings depending on the context of verb. Usually adding the word "up" changes the meaning slightly. However, there are some cases where difference is not that great or even optional.

Find something out is correct.

"Out" has similiar usage as well. Here's a useful video that explains how to use "Out" as phrasal verbs: http://www.youtube.com/watch?v=WsE4UmmkVtU

This website has a lot of great examples: http://www.phrasalverbdemon.com/particleup.htm

Hopefully this helps.

Which option is correct? and why?

"Drop her off" or "drop off her"
Pick him up or pick up him
find out something or find something out

My guess:

drop her off and pick him up.

because "him" and "her" are pronouns, if they are nouns like "the boy"

then both "pick the boy up"  and "pick up the boy" are ok.