I installed a 64-bit OS in a 32-bit processor

If you installed a 64-bit OS your CPU is necessarily 64-bit capable. In a 32-bit only processor the 64-bit installer not even starts.

In lscpu output CPU op-mode(s):32-bit, 64-bit means your CPU is both 32-bit and 64-bit capable. Architecture: x86_64 is the current kernel architecture (64-bit).

You can also check 64-bit support running:

grep " lm " /proc/cpuinfo

If it outputs nothing you have a 32-bit CPU. If it outputs something like flags : blah blah lm blah blah blah your CPU supports Long Mode (AKA 64-bit).

From your output it is clear that you have a 64bit CPU. The line CPU op-mode(s):32-bit, 64-bit means that you have a 64bit CPU.

Therefore there is no problem using a 64bit OS.

Looks like you experienced the same surprise I did a few years ago.

I accidentally put a 64-bit Ubuntu CD in my laptop and installed it, and a bit later I realised "Wait a moment.... I thought my laptop was a 32-bit system?"

If the 64-bit version works on your system, then that means your system is actually a 64-bit system, rather than a 32-bit one as you used to think ;)