Can I equip Skullclamp on a creature I am sacrificing?
It is possible to get the benefit of the creature's ability and the two-card draw from Skullclamp, but you have to do things in the right order and have a creature that can withstand a +1/-1.

There are a couple of rules concepts that are relevant here:


For the cards mentioned, sacrificing the creature is part of the cost.  You can tell because it appears before the colon (:)602.1a.  Paying a cost does not use the stack, so you can't "respond" to it.  As soon as you sacrifice the creature to pay the cost, it goes to the graveyard701.16a.
Creatures that have 0 toughness are put into the graveyard as a state-based action704.5f.  This condition is checked any time spell or ability resolves.


With this in mind, there are a couple of possible scenarios:


If you activate the creature's ability first, the creature is put in the graveyard as part of the cost.  This means you can't attempt to equip Skullclamp* while the creature's ability is on the stack - the creature is already gone.
If you Equip Skullclamp first, you have two options for timing your use of the creature's ability:


If you activate the creature's ability while Skullclamp's Equip is still on the stack, the creature is sacrificed before Skullclamp is equipped, so you won't get the two-card bonus.  (The Equip ability no longer has a valid target, so it fizzles.)
If you wait until after the Equip finishes, the result depends on how strong your creature is:


If the creature only has 1 toughness (like Diligent Farmhand), Skullclamp decreases its toughness to 0 and it will be put in the graveyard as a state-based action.  There is no opportunity to activate the creature's ability.
If the creature has more than 1 toughness, it will survive being Equipped with Skullclamp.  Its ability can then be activated, and you will get the two-card bonus when it is sacrificed.




So it is possible to pull off this combo with Dawntreader Elk, but not possible with Diligent Farmhand.**

*It's not generally possible to use the Equip ability like this anyway.  The Equip ability can only be used when you could cast a Sorcery.  There are ways around that, but they won't change this outcome.

**Unless you have some other way to boost its toughness or otherwise prevent Skullclamp from killing it.  There are always loopholes in Magic.