Process address space separation in L1 cache of Intel hyperthreaded CPUs
In modern Intel processors, the virtual-physical mapping is done in parallel with the lookup into the L1 cache.  There is a trick that gets the effect of doing lookups on virtual addresses, without actually having to complete the virtual-physical mapping before the cache lookup (and thus avoiding the delay you are worried about).  See https://stackoverflow.com/q/19039280/781723 and https://software.intel.com/en-us/articles/how-memory-is-accessedfor details of how they make this work.

There is only one L1 cache per physical core.  The two virtual cores on the same physical core share the same L1 cache.  See, e.g., https://stackoverflow.com/q/27797424/781723 and https://stackoverflow.com/q/32979067/781723.

Different processes can be hyperthreaded on the same physical core.

In general, you can often find answers to these questions by reading the Intel reference manuals.