Yes, there are decidable arithmetics. But Gödel's original 1931 proof was in the framework of Russell's Principia Mathematica, chosen because it was the most developed logical framework for reproducing all of mathematics at the time. Later he simplified the proof and showed that the Peano Arithmetic was enough.
Along the lines of your first suggestion, there is the Presburger arithmetic, introduced in 1929, about a year before Gödel obtained his result. It has addition, equality, and even induction, but no multiplication. The absence of multiplication precludes the definition of Gödel numbering, and therefore the construction of Gödel sentences. This is not a proof of completeness, but it explains why it holds retroactively, Presburger himself produced an explicit deciding algorithm. An algorithm it may be, but it is not too easy, Fischer and Rabin proved in 1974 that the computational complexity of the decision problem is doubly exponential. An interesting factoid concerning the difference between the two arithmetics: Glivický and Kala recently produced a model of Presburger arithmetic where the Fermat Last Theorem fails, and there are infinitely many counterexamples. It is believed that this can not happen for the Peano arithmetic, although this does not technically follow from Wiles's proof, see Are there non-standard counterexamples to the Fermat Last Theorem?
It is not so easy to implement the other two suggestions because finiteness is property of a model, not of a formal theory. Attempts to write finiteness into (first order) axioms, a la Dedekind for example, produce something which is not what it intuitively means (there are infinite models of theories which "claim" in axioms to be finite). If we settle for a model with finite domain instead of a first order theory though, that of course will be decidable, because one can check all possibilities by exhaustive search.
There is a kind of arithmetic with finite models, which preserves all the resources of the Peano arithmetic, but replaces the classical logic with the paraconsistent logic LP. These are Priest's inconsistent arithmetics. Note a subtlety in the notion of completeness for inconsistent models: while there is a decision procedure that assigns true/false to each sentence, there will be some to which it assigns both, called true contradictions or dialetheias. The reason we can have a finite model is that we can have "inconsistent integers" for which n=n+1, this leads to a true contradiction but does not trivialize the theory, paraconsistent that it is. In fact, for all numbers smaller than the least inconsistent number N the model is identical to the usual Peano arithmetic, and Priest argues that we can not possibly know what is "true" of objects in our world if N is exorbitantly large, see his What could the least inconsistent number be?
This vindicates Wittgenstein's paradoxical opinion that incoherence is not a problem for calculational mathematics (or any language game), and his anticipation of paraconsistent logics:
"Something tells me that a contradiction in the axioms of a system can't really do any harm until it is revealed. We think of a hidden contradiction as like a hidden illness which does harm even though (and perhaps precisely because) it doesn't show itself in an obvious way. But two rules in a game which in a particular instance contradict each other are perfectly in order until the case turns up, and it's only then that it becomes necessary to make a decision between them by a further rule... Well then, don't draw any conclusions from a contradiction; make that a rule."