32 vs 64 bit hardware and software

Hello folks,
this is the problem: At the company I am working we have some software
that has been developed on 32bit Linux. It works well, but its not
64bit ready. Unfortunately, new hardware is all 64bit. Changing the
source code of the application for 64bit would be quite time-consuming,
which means we want to postpone that. The following questions arise:
1.) Can I run 32bit Linux applications on a 64bit AMD/Intel hardware ?
2.) Can I run a 32bit Linux kernel on 64bit hardware *realiably* ? 3.)
Is there any way to run a 32bit binary on a 64bit kernel (or do I have
to go with 2. ?) ?
Thanks for your help !

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