Is suffering always present?
I think that the second noble truth says that suffering co-arises with craving -- and the third noble truth says that cessation (of craving and suffering) happens.

It's being called "pervasive", I think, comes from the tilakkhaṇa -- i.e.:


  
  sabbe saṅkhārā aniccā — "all saṅkhāras (conditioned things) are impermanent"
  sabbe saṅkhārā dukkhā — "all saṅkhāras are unsatisfactory"
  sabbe dhammā anattā — "all dharmas (conditioned or unconditioned things) are not self"
  


Note that it's "sabbe saṅkhārā" i.e. "all conditioned things" -- not "all dhammas" -- that are unsatsifactory.

Also note that "dukkhā" there is translated "unsatisfactory" not "suffering" -- i.e. the things themselves are unsatisfactory and impermanent.

Whereas "suffering" (with is another translation of "dukkhā", a context in which "dukkhā" is used) might be slightly different -- i.e. an aspect of consciousness which co-arises with craving and attachment -- i.e. craving and attaching to sankharas which are unsatisfactory and impermanent.