x86 live image on a x64 processor
Solution 1:
Yes it'll run.
A proper modern VM ought to run assembly language for the architectures in question with no issue.
Your VM software would have an option for running a 32 bit OS anyway.
In any case, x64 has support for pure x64 and combined 32 and 64 bit operations under long mode, and 16 and 32 bit operations under legacy mode. Your processor is probably frighteningly backward compatible.