Do the dice removed by cover and/or dodge count count as "Cancelling" for the purposes of suppression? That is, if an attack is negated by cover, are Suppression tokens still applied?

Suppression is applied as long as you roll a hit or crit on the attack. What happens after that roll is irrelevant.

16 hours ago, Reuben What Is? said:

Do the dice removed by cover and/or dodge count count as "Cancelling" for the purposes of suppression? That is, if an attack is negated by cover, are Suppression tokens still applied?

Correct, you would still receive suppression tokens. Suppression tokens are a result of hits/crits being rolled.

Think of it as your unit is under fire and they are scurrying for cover.