Yeah you can. But only the ones which haven't been updated and can be accessed using the 1.1.5 firmware. Apple is constantly updating their firmwares and apps follow. To have access for more apps, update your iPod. I'm not exactly sure if you can update an 1.1.5 but try clicking into the tab on iTunes where it lists your iPod under devices. It you can update it, it will say so in the update tab.
When you connect your iPod to iTunes, it should prompt you to install an update. After this you should be able to download apps.
If by that you mean it's a very old iOS, then the appmstore will prompt you to upgrade before you can download most things.
I believe so...but not the updated ones (I don't think) which are probably most of 'em. My best advice: update your iPod.
How? iTunes tells me the ipod is up to date...
So why not test downloading one...?
they won't let me download one...