Perfect secrecy with n-time key How can you encrypt $n$ messages with the same key, and have the same theoretical security you'd have encrypting a single message with a one time pad? For example, how can you encrypt two messages with the same key, and have that same theoretical security?

I think for $n=2$, something like $E(M) = (K_1M + K_0) \mod p$ should work, for prime p, $0 \leq M \leq p-1$, and each part of the key $K_i$ randomly chosen so $0 \leq K_0 \leq p-1$ and $1 \leq K_1 \leq p-1$.

This seems like the sort of thing that may have already been worked out or proved impossible, but I couldn't find it.
What you need for this is something called an $n$-wise independent hash function (like "pairwise independent" but $n$ instead). Such a hash function has the property that when applied to at most $n$ different inputs, its outputs are completely random. These can be constructed efficiently; e.g., a random polynomial of the appropriate degree works. What you actually defined with $E$ in your question is a linear polynomial with random coefficients and this is pairwise independent. There has been significant research on how to construct these, how to construct such functions that are almost independent, and so on. 

Having said all of the above, you don't really gain anything cryptographically with this, since the size of the key still needs to be larger than the size of everything you compute. So, you may as well use the one-time pad each time. However, there are other applications that such functions are useful (mainly in complexity, but they are also sometimes used in crypto).