Why does Kirk order Spock to initialise "Warp factor 1" in Dagger of the Mind? This order should be addressed to the helmsman, not the science officer/first officer, particularly so since Spock doesn't even have the helm the moment the order is given. Instead, he's at his post at the science station.
The responsibilities of the First Officer (FO) or Executive Officer (XO) include executing or having the personnel of the ship execute whatever orders the captain gives. Hence the captain is always free to either dispatch the orders himself, or to have the XO dispatch them as he/she sees fit.
That should be enough of an answer.
If you want something even further, we could hypothesize that "initializing warp factor one" involves preparations in the engine room (not present) as well as perhaps the helm, so it makes sense for the first officer to carry out the orders by dispatching them to the various parties appropriately.
Just to provide some references, in Wikipedia, First Officer redirects to Executive Officer which is defined as
An executive officer (XO) is generally a person responsible for running an organization [...] . In many militaries, an executive officer is the second-in-command, reporting to the commanding officer. The XO is typically responsible for the management of day-to-day activities, freeing the commander to concentrate on strategy and planning the unit's next move.
and under the U.S. Navy section (on which we know Starfleet is based):
An XO is assigned to all ships, [...] and is responsible to the captain for all ship’s work, drills, exercises, personnel organization, and the policing and inspection of the ship.
Just as additional color: We can see how this works in "Chain of Command" where Capt. Jellico takes over command of the Enteprise and then issues high-level orders for the XO (Riker) to execute. When he does not execute them in a timely fashion, Jellico is taken aback since (in his view) the XO's responsibility to get it done:
JELLICO: How many duty watches does the crew stand?
RIKER: We've a standard three shift rotation.
JELLICO: I'd like to change that to four starting tonight. I'd also like to examine the duty roster and the crew evaluations as soon as possible. I want readiness reports from each department head by fourteen hundred hours, and a meeting of the senior staff at fifteen hundred.
JELLICO: Is there a problem with delta shift, Will?
RIKER: There is no delta shift yet, sir. I have spoken to the department heads about changing from three shifts to four, and they assure me it's going to cause us significant personnel problems.
JELLICO: So you have not changed the watch rotation.
RIKER: I was going to explain this to you after the ceremony, sir.
JELLICO: You will tell the department heads that as of now the Enterprise is on a four shift rotation. I don't want to talk about it. Get it done. Now that means delta shift will be due to come on duty in two hours. I expect you to have it fully manned and ready when it does. Is that clear?
RIKER: Yes, sir. If you'll excuse me, sir. Captain.
JELLICO (to PICARD): He was your first officer for five years.
The DS9 Episode For the Uniform, implies that there are various communications between the bridge and other departments that happen more or less automatically on a fairly regular basis
O'BRIEN: With most of the bridge control functions offline, all orders to Engineering have to be relayed.
This means that, for example, when Picard orders the helm to set a course at a given speed and engage, orders go to engineering to travel at warp X.
There is also evidence in Chain of Command to suggest that it's not unheard of for Captains to relay orders to lower ranking officers via other command staff.
JELLICO: Data, I want to be at Minos Korva in one hour.
DATA: Aye, sir. Set course three five zero mark two one five and engage at warp eight point five.
In this case, Jellico was on the bridge and within earshot of the helmsman, so he could have easily given the order himself. That said, he may not have known the correct heading or the speed needed to arrive in one hour.
So why does Kirk give the order to Spock instead of doing it himself?
He had been through a lot in that episode. It's possible, he didn't remember (in the moment) what their next mission was or that he just didn't want to deal with all of the associated orders he'd have to give with the speed order. Lucky for Kirk, he has a ship full of underlings to do stuff for him.
Probably because the actors' contracts specify a certain number of lines per episode. Having Spock reply to Kirk's order satisfies that requirement.