Let $G$ be a compact group and $K$ a finite extension of $Q_{p}$. If $\rho$ is a continuous representation of $G$ on a finite dimensional vector space over $K$, then it is well known that the semisimplification of the reduction modulo $p$ of an integral model of $\rho$ only depends upon the isomorphism class over $K$ of $\rho$. Are there results or strategies that, given a $K$representation $\rho$ as above, allow one to determine properties of all the possible integral models of $\rho$? For example: are there some conditions under which all the integral models of $\rho$ are semisimple? Where can I find references? Thanks!

Let be focus on the twodimensional case first. Finding an integral model for $\rho$ is equivalent to choosing a lattice in $K^2$ that is invariant under $\rho(G)$. So in this case you are asking for invariant points in the tree for $GL_2(K)$ under a compact subgroup $\rho(G)$ of $GL_2(K)$. There are various possibilities, and there is a nice paper of Bellaiche and Chenevier Sousgroupes de $GL_2$ et arbres discussing the possibilites. When you say that you would like all integral models to be semisimple, I'm not sure what you mean; but it say the represenation over $K$ is irreducible, then it will never happen that all the integral models have semisimple reduction, if they are not themselves irreducible. (This is a proposition of Ribet, and is easily seen in the treebased picture.) Just to illustrate how one argues: if two points in the tree are $\rho(G)$invariant, then so will be all the points on the line segment joining them. If $\rho(G)$ fixes an infinite halfline, this corresponds to an invariant onedimensional space over $K$, and so means that $\rho$ is reducible over $K$. So if $\rho(G)$ is irreducible over $K$, then the fixed set of $\rho(G)$ will be bounded and convex, which already imposes some restrictions. (One an prove Ribet's result by these sort of considerations: basically, any extremal point of the fixed set of $\rho(G)$ will have nonsimple reduction.) Once you understand the $GL_2$ case you can imagine how to try and generalize things to the $GL_n$ case, although the combinatorics becomes more complictated, and (as Bellaiche and Chenevier note) it is harder to say anything nice, because when $n > 2$, $GL_n$ is the full automorphism group of the building. 


For general representations, there are no such strategies and this is a very hard open problem, even for finite groups. Since your usual compact group will have various finite quotients, your problem is at least as hard. This rather older survey by Reiner is a good place to start. Particularly pages 171173 should be of interest to you. As for semisimplicity, you need to be clear about what you mean. In the integral world, the classical notion of semisimplicity makes little sense, because your representations always have subrepresentations. E.g. if $\Gamma$ is an integral model, then $p\Gamma$ is a subrepresentation. But at least the KrullSchmidt theorem holds in your setting, even though the indecomposable summands of an integral model might not sit in irreducible $K$representations. In fact, if the semisimplification of the reduction of $\rho$ contains two nonisomorphic irreducible components, then I believe that there is almost no hope for all integral models to only have indecomposable summands that sit in irreducible $K$representations. (Clearly, if the reduction itself is not semisimple then your integral models will be at least as badly behaved.) 

