I'm worried about my Windows apps modifying the files in Ubuntu, as there is software in Windows that can see and can modify files on the Linux partition. Can files or applications run normally in Ubuntu after modifying? If so, how can I watch the files for changes in Ubuntu?
In general, yes, files on Ubuntu will be fine if they're modified in a way they're intended to be, even though it's from Windows. For example, editing an image with Photoshop.
However, there are a bunch of little exceptions, and the only way to find out for sure is to experiment:
The safety of your Linux partition will depend on the quality of the Windows filesystem driver: will it correctly use the journaling? Adjust the timestamps appropriately? Will it lose data, or is it rock solid? Etc. Personally, I'd look into exactly who made the Windows software which will be used to mount the Linux partitions.
Some file types are platform independent (e.g. png), but some are not (e.g. text). If you edit a Linux text file in Wordpad, you may find that every line looks odd because line endings are different in the two systems.
There are other odd exceptions such as Linux filenames always being case sensitive, while this is not the case for all Windows filesystems. And so small little oddities can creep up.
Most of these issues aren't tragic and can be easily corrected or worked around. But they're things to keep in mind.
My advice: don't go through the hassle of multi-boot situations. Just get a second computer (a cheap one if necessary) to run the other OS on. Or, boot another OS in a virtual machine.