Running 32 bit app on a 64 bit Linux machine

From Wikipedia: “In 2004, the initial 32-bit x86 instruction set of the Pentium 4 microprocessors was extended by the 64-bit x86-64 set.”

Your /proc/cpuinfo output shows flags: … lm …. The flag lm stands for “long mode“ which means 64-Bit extension. Thus, you have indeed a 64-bit processor.

32-bit x86 CPUs (since the Pentuim Pro) support up to 64 GiB RAM (using PAE). (The "CONFIG_HIGHMEM64G" kernel option needs to be set to actually use it). Each application can only see 4 GiB at a time (and some of that 4GiB must be used for other things, the exact amount depending on the "Memory split" kernel setting)

64-bit operating systems have some other advantages as well, such as access to extra registers on the CPU, which can speed up some types of applications (by allowing more temporary data to be kept in the much faster registers, rather than main RAM)