You (almost) always put always before the verb because adverbs of frequency precede the main verb. There are always(!) exceptions that proves the rule.
Same applies to specifically. You'd usually put it precedent to the main verb but there are reasons to move it in end position which I will discuss shortly in the long answer.
I always confuse...
... they specifically target us...
... they target us specifically.
There are different kind of adverbs and there are different "rules" where to put them. Adverbs can take the initial, mid and end position.
Without going too deep into detail, here are some ideas on how to decide where to put the adverb. Let's start with adverbs (of manner) that modify a single word.
Broadly speaking, the adverb is preceding the word it's modifying. If the adverb modifies a verb, you place it before the verb. In the following sentence, for example, the adverb carefully modifies the verb to drive.
The man carefully drives the car.
If you want to put focus on how something is done, you can move the adverb to the end of the sentence.
The man drives the car carefully.
If the adverb modifies an adjective, you place it before the adjective. If the adverb modifies another adverb, you place it before the adverb. Those adverbs usually specify the certainty and degree of something. Here's an example for both:
The very old man drives the car extremely carefully.
The adverbs very and extremely modify the adjective old and the adverb carefully, respectively.
Adverbs of indefinite frequency (as always) are used likewise, i.e. they are in a mid position.
I always forget where to put the adverb.
Note, that an adverb is always after an auxiliary verb.
I have always been bad at remembering the position of adverbs.
Adverbs of place, time and definite frequency usually go in end position:
I learned English in school.
I saw her last week.
Again, in order to put emphasis on the adverb, you can move the adverb to the front.
The weather is still fine, but it will rain tomorrow.
The weather is still fine, but tomorrow it will rain.
Eventually, a linking adverb can take the initial position but also the position precedent to the verb. If you are unsure about this, simply take the initial position. Same applies to adverbs that have a commenting function or determine the viewpoint.
Officially, I am not allowed to tell that.
Generally speaking, learning English is fun.
However, this does not apply to me.
With this in my mind, a grammatical version of your sentence is
Why did they specifically target us?
Most natural in this situation, however, is to put the adverb at the end of the sentence as it put focus on that you care about "why us and not others".
Why did they target us specifically?
There is another alternative if you want to know the specific reason for targeting us:
Why, specifically, did they target us?