Suppose you are given a closed subvariety $V$ of projective space $\mathbb{P}^n_k$. Let's say we fix the degree and the dimension of $V$. Can we then bound the arithmetic genus of $V$, or does no such bound exist?

I don't know about an explicit bound, but a bound exists in theory for arbitrary varieties. This follows from the fact that the set of cycles, and in particular subvarieties, in $\mathbb{P}^n$ of fixed degree $d$ and dimension $N$ are parameterized by a Chow variety $Chow_{d,N}$ (it needn't be irreducible, but it is certainly of finite type). More formally, consider the preimage of $Chow_{d,N}$ in the Hilbert scheme. This preimage has finitely many components, and therefore there are finite number of possible Hilbert polynomials for a fixed $d,N$. This is a bit sketchy, but perhaps someone else can supply more details or precise references. 


I believe it was first proved by Kleiman (you only need to assume fixed dimension and degree, no need to assume a subvariety of some fixed $\mathbb P^n$), see Corollary 6.11 S. Kleiman, Exp XIII in A. Grothendieck et al., Theorie des Intersections et Theoreme de RiemannRoch (SGA 6), Lecture Notes in Math No. 225, SpringerVerlag, Heidelberg (1971). 

