Do you think a dice roll is random?

It is in the sense of being unpredictable (assuming person not deliberately cheating). There certainly some aspect of rolling that may reduce randomness (unevenly balanced dice) but as long they are not easily predicted, its random enough for many usecases.

Random though: First thing that came to my mind at school when I was introduced to theory of probability was "if you really sit down and start rolling a dice hundred of times to see how random it is... what about those times when a dice will fall into crack in the floor, your cat will take it over or otherwise unmeasurable results? The answer "we ignore those" made me realize that I am to practical for math :-)

Assuming perfectly balance dice on a uniform, moderate friction surface, rolled after being shaken, I would be guessing - probably. Would be interesting to roll one 600 times and see how close to each number comes to appearing 100 times.

How about a DIY project where you build a gadget (robot?) that rolls the dice, records the result and repeats it 60,000 times ? And that is to test a single dice. Why 60,000 ? because if the machine could do one test each second, then it would take 17 hours for a result. Any more than that and most of us would be getting impatient.

One more answer for this question: randomness is measurable. There are tests for it, though (the ones I've heard about) were mostly practical (check for certein set of patterns) rather than mathematical proofs.

I doubt quantum indeterminism takes place at the scale of a dice

So, no

"dice random test"