If Linux is installed, which is what you're doing, it's not "live," regardless of the media that you install it to. Live Linux (with or without persistence) is when you boot your computer from the ISO.
You can install 32 bit applications to a 64 bit system.
To run a 32-bit executable file on a 64-bit multi-architecture Ubuntu system, you have to add the i386 architecture and install the three library packages libc6:i386, libncurses5:i386, and libstdc++6:i386
Further details here:http://askubuntu.com/questions/454253/how-to-run-32-bit-app-in-ubuntu-64-bit
Personally I would do what (I think) Nick is suggesting, and install both 32 and 64 bit Debian systems on separate partitions on the USB. That is, if installation is best for the purpose.
Are you sure that this USB will work on all the computers you want to use it on? When you use a computer to install Linux (whether to a HDD, USB or whatever), it will choose drivers based on the hardware on that particular computer, and may or may not work on other computers.
Using Linux Live with persistence however,
1) you can install applications, and
2) the ISO performs checks on the hardware every time you plug the USB into a computer to boot it, in order to select appropriate drivers for the hardware it's being asked to use. Whereas if you install Linux to the USB, it will expect to use whatever hardware is on the computer that you used for the installation.
Personally I prefer installing Linux to USB rather than use live with persistence, but as you need to use it on different computers perhaps live with persistence would be best.
Edited by Al1000, 16 December 2015 - 03:55 PM.