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Your last statement is not true in general. Let $G=C_3$ and take your favourite finite field that does not contain the cube roots of unity, e.g. $K=\mathbb{F}_5$. Then the two non-trivial one-dimensional representations over $\bar{K}$ are not defined over $K$, but their sum is, since it's the regular representation minus the trivial.
In general, $K[G]$-modules for $K$ finite of characteristic co-prime to $|G|$ behave pretty much like modules over characteristic zero fields (the simple $K[G]$-modules are just sums over Galois orbits of the absolutely simple ones), the major simplification being that there are no Schur indices. I am not sure that there is much more to say about this. The second volume of Curtis and Reiner contains a whole chapter on rationality questions, i.e. fields that are not "sufficiently large" in the sense of Serre. Most of it is for characteristic zero, but as I say, a lot of it carries over.
Edit Re updated question: if $G$ is a finite $p$-group and $K$ is a finite field of characteristic different from $p$, then it is indeed true that any irreducible representation of $G$ over $K$ has dimension $tp^k$, for some $k$ and some $t\;|\;(p-1)$. Indeed, the absolutely irreducible representations have dimension a power of $p$, and the field of definition of any absolutely irreducible representation is $K$ adjoin $p^r$-th roots of unity for some $r$, so the Galois orbit of a representation has size dividing $(p-1)p^{r-1}$ for some $r$.
Your last statement is not true in general. Let $G=C_3$ and take your favourite finite field that does not contain the cube roots of unity, e.g. $K=\mathbb{F}_5$. Then the two non-trivial one-dimensional representations over $\bar{K}$ are not defined over $K$, but their sum is, since it's the regular representation minus the trivial.
In general, $K[G]$-modules for $K$ finite of characteristic co-prime to $|G|$ behave pretty much like modules over characteristic zero fields (the simple $K[G]$-modules are just sums over Galois orbits of the absolutely simple ones), the major simplification being that there are no Schur indices. I am not sure that there is much more to say about this. The second volume of Curtis and Reiner contains a whole chapter on rationality questions, i.e. fields that are not "sufficiently large" in the sense of Serre. Most of it is for characteristic zero, but as I say, a lot of it carries over.