What post-quantum assymetric crypto is there?

Wikipedia to the rescue: https://en.wikipedia.org/wiki/Post-quantum_cryptography. My personal favorite is the McEliece cryptosystem, based on error-correcting codes: https://en.wikipedia.org/wiki/McEliece_cryptosystem. They key size is huge (well, under 1 MB still) but computation isn't too bad. I'd still recommend adding RSA plus several post-quantum schemes in an XOR chain as I described.

About increasing key size without a clear need, a lot of crypto algorithms take compute time that grows faster than linearly with key size. Executing several independent algorithms in parallel is better for two reasons: first, the key sizes of each one aren't large so don't suffer the nonlinear slowing, and second, they can be executed on separate cores in parallel.

I'd welcome advice from an expert, but my impression is that the mainstream crypto researchers think that it's more conservative to adopt a single, trusted crypto algorithm and bet the farm on it. My instincts are that this is a bad approach. Composed algorithms like the one I described where all of (say) 5 schemes must be cracked before the attacker gets anywhere are more conservative in my view since they are at least as strong as each of their constituents. However, I'm not a crypto researcher, and there might be a good reason not to shield RSA (which we know is secure to classical but not quantum attacks) with a variety of layers that each provide a good chance of being robust against a quantum attack.

When will we have quantum computers? One reasonable scenario is that by 2020 we'll have a Sputnik moment where somebody will build a quantum computer much better than the sleepy mainstream expects, yet not powerful enough to run Shore's algorithm against 1024-bit RSA. This will shock the world into a bit of a panic that a bigger quantum computer will come soon, and RSA and elliptic curves will be seen as untrustworthy by 2025. We'd be better off adding a layer of protection now, especially since we're sending data now that we wouldn't want to be public for a lot longer than 2025.