Can a non-rotating ball have angular momentum?

Yes, it does. It may seem a bit more intuitive if you imagine a line connecting your reference point and the centre of the ball: as the ball moves, the line sweeps out an angle across the ball, so there should be some angular momentum. If you think in polar coordinates instead of cartesian, the ball is rotating.

I think your question comes about because you are comparing angular momentum to linear and there seems to be a difference: in the case of "normal" momentum, it's mass times velocity, it's a "real" thing... but angular, well that seems to be virtual, the object might have or not have it depending on where "you" are. And that seems strange, right?

Ok, so imagine that same ball going along that straight line, but now you are on a cart travelling at the same speed along the same line. Now what's the momentum of the ball? Zero!

This example doesn't seem so odd, but the underlying basis is the same; all of these measurements depend on what "you" are doing, or more technically, your frame of reference.

So the idea that an object can have multiple values for [MEASUREMENT X] at the same time should not be at all surprising or weird, it's kind of baked into the way these things are defined.

Yes. If you watch a vehicle going straight on road while you stand at a bus stand then the angular momentum possessed by the vehicle is $m\vec{v}\times\vec{r}$, where $m$ is mass of vehicle, $\vec{v}$ is velocity of vehicle and $\vec{r}$ is the vector starting from you and ending at the vehicle which is variable in this case.