If I have Breya, Etherium Shaper and two Thopter Tokens as well as Skullclamp
and I attach a Skullclamp to the Thopter Token and activate one of Breya, Etherium Shaper's abilities in response to the thopter being equipped, do I draw two cards and get the benefits of one of her abilities?
No. Once the equip ability resolves the Thopter will be a 2/0 and placed in the GY immediately and skullclamps ability will trigger. If you were to have an anthem of some sort (Intangible Virtue) then the creature would instead be a 3/1 and if you were to then sacrifice it you would get both the Skullclamp trigger and the ability from Breya
If you sacrifice the token targeted by Skullclamp's equip ability before it resolves, Skullclamp will not be attached to it (and won't be an "equipped creature") so you won't get to draw two cards from Skullclamp's triggered ability. On the other hand, if Skullclamp becomes attached to that token and the token's toughness becomes 0 or less as a result, the token will go to the graveyard as a state-based action before any player gets priority to activate any abilities, including Breya's activated ability (C.R. 704.5f, 116.5) (but even so, Skullclamp's ability will trigger, since it doesn't matter how the token died). (Note that Breya has only one activated ability, not several; you choose which mode to use when you activate that ability [C.R. 601.2b].)
and I attach a Skullclamp to the Thopter Token and activate one of Breya, Etherium Shaper's abilities in response to the thopter being equipped, do I draw two cards and get the benefits of one of her abilities?
EDIT (Dec. 15): Added rule citations.
EDIT (Jun. 10, 2019): Clarification.