sentence structure: location of object after action applied to it
1 and 2 are correct, but 2 is weird.
You can find a nice explanation here.

The grammatical role of 在桌子上 is sightly different though.
In: 


  他把书放在桌子上。


The phrase 在桌子上 is a result complement. The verb 在 acts as a result of 放：the result of 放 "putting" the book is that it now 在 "stays" on the table.
Now the result verb must occur right after the main verb, therefore the object of the action gets pulled in front of it by 把， thus achieving a SOV structure.

In example 2:


  他在桌子上放书。


The phrase 在桌子上 acts as a location complement, and therefore locates the action 放。The sentence is weird because it's not clear if being on the table is the result of the action or where the action is taking place.

Examples 3 and 4 are not grammatical.

So to answer your question, anticipation of the object with Subj. + 把 + Obj. + Verb Phrase. happens mainly in two cases:


when the main verb is followed by a result verb or a result complement 


  你把苹果吃掉吧。 Eat the apple. (result is 掉 = object has disappeared)
  你把钱包拿出来。Take out the purse. (result is the purse comes out)

when the main verb is ditransitive (direct and indirect object)


  我准备把这件礼物 (direct obj.) 给 他 (indirect obj.) I'm going to give this present to him