It's possible to run Ubuntu straight from a USB stick and use the same stick as HDD?
Yes it is possible. when you install the OS make sure to install the OS in the USB drive of your choosing.
1) when you install the OS go to something else option
2) select the drive that you want to install the OS (In your case your USB drive)
3) select USB drive as root partition (/)
4) after that change boot-loader install location to your USB Drive
5) click continue
now if the installation successful then your USB drive is now bootable.
and don't forget to override the boot order inside the BIOS when you want to use that USB drive.
What you describe is called persistent live installation. You can achieve this in two ways:
- favourite option: https://unetbootin.github.io/ (you can also partition the USB storage in two parts with gedit
- alternative option: create a casper-rw "virtual partition" (it's nothing but a file in fact) for the persistent data storage. (see How do I get a live-USB to use a partition for persistence?)