Yes, the official CPU requirement for Windows 7 32-bit is simply a x86 CPU clocked at 1 Ghz or above.
Any Intel Pentium 4 meets those technical specifications.
About Windows 7 64-bit, it depends on model of Pentium 4 : first models don't support x86-64 instructions but the newer models fully support them.
Nowadays, a lot of netbooks with single core Atom are shipped with Windows 7 despite the fact that power processing of these netbooks is significantly below most Pentium 4-based system (especially Pentium 4 with HyperThreading)
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Yes. However, you must have at least 1 GB of RAM, which is uncommon in most Pentium III computers, even when fully upgraded.