Every time I move my right stick upwards, I do an ollie. No flick.

its the same for me, i can do any trick which my stick moves downwards, but, moving my stick upwards causes me to ollie preventing me from doing anything else

I found a fix for me! Hopefully it can help others!

I almost gave up playing it. My problem sounds similar and I bought a second controller because I thought it was just a busted controller stick. I was diagnosing it with the "flick visualizer" on in the game. With the visualizer up, I would see the dot sit at the bottom left corner until I pushed any button or moved any stick. This meant that anytime I was off board the camera would spin in circles. Also in menu the character would always spin or the selection would constantly scroll. And also I couldn't nollie because any forward motion would cause a small ollie and the characters legs would spin to land crooked. I figured out that I could turn the right-stick-click button to a toggle switch so I could trick the game into thinking I am always holding a button. Right-stick-click was chosen because there were no in-game settings tied to it that I could find.

Anyway, here was my fix on PC:

1. Go to steam library
2. right-click on Skate.
3. select "properties"
4. select "controller"
5. At the top of the window there should be a link to "controller configurator", select it
6. select "edit layout"
7. select "joysticks"
8. Next to "right stick click" there should be a gear icon, select it
9. select "settings"
10. At the bottom of the list I have "toggle" selected and "interruptable" selected.
11. Done, close out of the pop-ups.
12. When you get in game, it will still spin. Click your right stick once and for me it permanently stops!

Update;

I learned that right-click is something with interacting with other users in the server. So anytime you skate near someone it opens a dialog box about who they are. I have been playing just dodging people or toggling on-off right-click when I need to be by someone. It is a 95% fix. It definitely makes the playing experience better but not a full fix.

FIXED.... Gents simply disconnected any Joystick, steering wheel or pedals USBs from PC or console. 

what control settings have you got on? afaik need your settings on classic controls if your on streamlined cant do tricks nollie