You can install either 32-bit applications or 64-bit applications. Both will run without any problem, but you should experience a performance boost with 64-bit software.
32-bit applications will default to Program Files (x86) whilst 64-bit applications will default to Program Files.
You will find two IE9 applications: one will be the 32-bit version and the other will be the 64-bit version.
As for Adobe Flash, the latest 64-bit version includes the 32-bit version as well.
Also, be aware that older 16-bit software will not run on a 64-bit operating system unless you use something like Virtual PC and XP Mode, for which you would at least need Windows 7 Professional.
To recap, if you have any 64-bit software, don't hesitate to install it.