Grammar – Why Is ‘Tomorrow It Will Have Been a Year Since We Came’ Incorrect?

"I will have been 7 years in the country this October"

...is fine, but I'd rejig it a bit, so:

"I will have been in the country seven years this October".

For the second, you could say:

"Last October was the start of my eighth year in the country".

(1) I must have finished this exercise by tomorrow.
(2) He must have dropped the pen by mistake.

Note, first, that you are dealing here with two different senses of must. In (1), must has the deontic sense of obligation: you are required to have finished the exercise by a particular time. In (2), must has the epistemic sense of inferential necessity: you conclude that he dropped the pen by mistake.

(1) is an acceptable alternative to this:

(1a): I must finish this exercise by tomorrow.

The two versions take different perspectives. In (1a) you are speaking of an obligatory task which now, in the present, lies before you; in (1) you are speaking of an obligatory state, that of having finished the task, which will obtain in the future.

But although (1) is grammatically acceptable, most speakers would not use it to express that future obligatory state. Instead they would employ a very similar but not identical idiom:

(1b) I must have this exercise finished by tomorrow.

(1b) deploys the participle finished as an adjective modifying exercise: your obligation is seen not as that of finishing the exercise but of presenting the exercise in a finished state.