Is the following extension of finite state automata studied? Consider a finite state machine as usual, but every transition, it can also update an integer counter by adding or subtracting a number.
Say, a transition function of the form
$\delta(q,a) = (p,k)$
moves to the new state $p$, and add $k$ to the counter, where $k \in \mathbb{Z}$ (so $k$ can be positive, negative, or zero).

A string is accepted if the final state and the counter value is in $F$, where $F$ is a finite set of pairs of states and counter values. 

Is this model known? I could not find any reference of this particular extension.
Assuming $k$ can be any integer, then this can be formalized as a blind one-counter automaton. Usually these automata accept on final state when its counter is zero, but we can easily model your acceptance type if you allow $\epsilon$ transitions (that do not consume input). If I am not mistaken, like with finite state automata, one can get rid of the $\epsilon$, but that is a non-trivial result.

There are several types of one-counter automata. In the most general form they are allowed to test whether the value of the counter equals zero. 
The languages they accept are a strict subset of the context-free languages. 

The model you are probably looking for is called blind, it cannot test for zero, except as the final test for acceptance at the end of the computation.