I will (have the package sent) vs (have sent the package) by next week

I will have (past participle of verb) (object) is English's "Future Perfect" tense construction. It refers to actions (on the object) that you will complete before a certain time in the future.

I will have (object) (past participle of verb) is a construction that in English means "I will direct that the object be (verb-ed)".

I will have the package sent by next week

is acceptable, and means that before next week, you will have arranged for the sending of the package.

I will have sent the package by next week

is also acceptable, and means that before next week, you yourself will have done the sending of the package.

By next week I will have discussed the project with the professor and would be able to answer your questions then.

is acceptable and means that before next week you and the professor will discuss the project.

By next week I will have the project discussed with the professor and would be able to answer your question then.

is unusual. It might mean that before next week, you arrange for the professor to discuss it with someone else (not you).

This is a puzzling set of constructions, and it's puzzled many people before you.

To start with, all of the sentences you cite are grammatical, which means there is nothing about "correctness" here.

As for the constructions themselves, one

• I will have sent the package by next week

is a normal Verb Phrase construction with three component verbs:

• a Modal Auxiliary (will), which requires next
• an infinitive of the next verb in the VP chain (have), which requires next
• the past participle of the main verb in the VP (sent).

Some would call it "The Future Perfect Tense", but it has exactly the same structure as

• She may have lost her keys.
• He couldn't have done it without you.

which some would not call "The Future Perfect Tense", although they may not know what other tense to call it.

In the second construction

I will have the package sent by next week

the participle sent is no longer the main verb, but has a different function in each of the three constructions, all idiomatic, all with the identical syntactic structure

Subject (+ Aux ..) + have + NP + Participle

(in these constructions, get can be substituted for have at will in American English)

One is the construction Mark described, which requires an Agent subject, someone to be responsible for getting somebody to get the job done.

Another is a construction with a Patient subject which refers to having something happen to the subject, typically something unpleasant, and typically without the subject's volition:

• He might/will have/get his tires slashed if he parks there.

And the third construction, this one neutral about pleasantness, means 'to put NP into a Participle state':

• We might have the car washed by the time they get back.

This third construction is the one that happens to wind up meaning pretty much the same thing as the Perfect construction, even though it gets there by a different route. If it is true that you will have put the package into a "sent" condition by next week, then it is true that you will have sent the package by next week. And vice versa. This sort of thing happens all the time in a language with so much syntax, like English.

But they're all grammatical.