Can a ball stay still while laying on a inclined plane?
In freshmen physics, the inclined plane and ball are perfect and the ball moves, so for your purposes, no.
If either the surface or the ball have imperfections, we can tip the plane and the ball won't move until gravity exceeds the sum of the normal forces. To imagine those normal forces, we look very closely at the interface between the ball and the surface to see that the ball rests on high (3+ non-colinear) points on the surface. So long as the high points (of the interface) surrounding these three points are similarly high, the ball rolls when the gravity vector points outside that triangle.
I'm wondering: if friction force is strong enough to counteract the component of gravity force parallel to the plane, will the ball even start to roll/slide?
The ball is either moving or the imperfections in the ball and plane are keeping the static condition outlined above. In the latter case, the normal force is stopping the ball, not the friction force. Imagine a bicycle with one wheel on a flat stair and another on a higher or lower flat stair - will the bicycle accelerate?
Well, since the net force acting on the ball is zero, I think the ball would not roll nor slide down. It won't be totally still, though. There is the torque from the friction force, which would make the ball skid in its place. Does that make any sense?
It sounds like you want the ball to start spinning without translating. No, that doesn't make any sense.
To me it is the same case of placing a spool on an inclined plane while holding its string.
Holding the string gives you a no slip condition on one side and either static (no movement) or kinetic (it translates and rotates) friction on the other.
As the gravity force tries to push the spool downwards the plane, I pull the string to counteract the gravity force. The spool stays on its place, just rotating (on this case, I mimic the friction force by pulling the spool string upwards).
If you pull on the string, you're doing something more complicated and un-sustainable as your arm isn't as long as the string.
Even if my analysis is correct, I think no practical surface would have a friction coefficient high enough to keep a ball from rolling and/or sliding.
Your analysis is not correct. Velcro and glue do a nice job of arresting movement.
To solve problems like this, check to see if the ball rolls by using the static friction inequality and then use either a no-slip condition or a kinetic friction force that appears in both the the sum of the forces and the sum of the torques. One might also choose to use the energy equation for the no-slip case.