In general, no. Linux understands windows and other OS file systems just
fine, but windows has no clue about non-windows file systems.

However, there are some hacks written which allow a windows pc to see a
subset of the availble linux file systems (ext2/ext3 only, I believe)
but in general it's a very, very bad idea, security wise, to give a
windows pc the power to scribble all over your linux filesystem.

If you're determined to put your linux system at risk though, just look
for the ext2 filesystem driver for windows, and knock yourself out.

Yes, you can copy all of them.
You can even do the following: copy all you have in partition C to your Ubuntu
partition in a folder in order to preserve all you have under Windows, including
installed programs etc.
Then, when your Windows installation gets messed up in a few months or years
you can just delete everything you have on partition C and copy back your
preserved Windows installation. Such, you will not have to reinstall. But it
occupies space.
I have done this with Windows XP and it worked fine, except for that, at
every startup it showed me some text in Notepad with some info I did not
understand.
So, a Windows installation can be preserved under Linux.
Another funny thing would be to delete your System32 folder in folder Windows
and then try to boot Windows. In this case I also recommend you to have a
preserved System32 folder which you can slam back into the Windows folder.
Another funny thing is starting IE or Notepad in Wine.

I agree with Alex. Windows files are easily copied to the Linux desktop.
Whether you can use them depends on what type of file it is and how much
Linux support there is for it.

My preferred method is drag and drop from the file manager. Windows drives
or partitions are detected and mounted automatically, although you may be
asked for a password to mount it after you click on it (or not... depends on
the version and distribution). Windows standards work here. If you press the
shift as you drop it will move, instead of copy. Ctrl+A for select all and
holding the Ctrl key while selecting multiple files work. You can also copy
(or cut) and paste, if you prefer.

Yes. When I open "computer" one of the things which appears in the column on the
left is "62.8 GB Media". That's my Windows 7 C: drive. I can click on it, then
double-click my way through the folder structure, eventually selecting a file to
copy. Another item in the column on the left is "Desktop". I can click on it,
and select "paste" and I have just copied a file from the Windows drive to my
Linux desktop.