Homomorphic cryptosystems in RSA
Yes, homomorphic encryption operations are malleable by definition. The definition of "malleable" is something along the lines of "can be intelligently modified", which is what homomorphic encryption allows you to do.


  That you can preform operations on two unknown (encrypted) ciphertexts, $\epsilon(x_1)$ and $\epsilon(x_2)$, and recieve the result, $\epsilon(x_1\cdot x_2)$ by preforming $x_1^ex_2^e\bmod m$? Or is it just showing that $\epsilon(x_1)\cdot \epsilon(x_2)$ is the same as   $x_1^ex_2^e\bmod m$?


Those are the same thing by definition. The encryption operation will allow for the multiplication of two ciphertexts to equal the encryption of the multiplication of the two plaintexts (mod $m$, in your case).

And yes, you would use the RSA private key for decryption. The public key would contain the exponent e in your equation.

Edit, for completeness:

Note that textbook RSA usage (aka, simply taking the plaintext to an exponent) is not secure for several reasons. When security is necessary, we apply a padding scheme to the plaintext before encrypting it (aka, $m$ becomes $p(m)$ for some padding function $p$). But doing so causes the homomorphic usage to multiply the formatted plaintexts instead of the actual original plaintexts (aka, you wind up encrypting $p(m_1) * p(m_2)$ instead of $m_1 * m_2$), which won't yield the desired results. The padding schemes will completely mess up the ability to use homomorphic properties, since multiplying formatted texts yields garbage.

You mention this is for a game, so security may not be of utmost importance (it depends on what the stakes are and what data is involved), but take care before using RSA without any padding; if you do use it, investigate ways to mitigate at least some of the weaknesses of textbook RSA.