Power Dissipation VS Switching Power
Must be folks are out at the bar, tonight. I'd have figured lots of answers showing up for this one in less than an hour. Nothing. So I'll provide my hobbyist view.

The datasheet can be found here: AO3400 N-Channel MOSFET. It specifies \$R_{DS(ON)}\lt 28 \:\textrm{m}\Omega\$ at \$V_{GS}=10\:\textrm{V}\$. Since you are using \$12\:\textrm{V}\$, it's probably fair to assume that they have access to a \$V_{GS}\$ of that level. You also mention a mounted \$10\:\textrm{W}\$ LED. So from this, I can presume that the current is \$\le 1\:\textrm{A}\$. It follows that the power dissipated in the NFET, based upon \$I^2R\$, is about \$30\:\textrm{mW}\$. The maximum thermal resistance to ambient is \$125\:\frac{^\circ \textrm{C}}{\textrm{W}}\$, so I guess we are talking \$\approx 4\:^\circ\$C over ambient here?

(It's possible that it is well-connected thermally to the board, as well. And perhaps closer to the typical thermal resistance than the maximum. So it's probably a fair bit less than that.)

That answers your non-question, "it was not even warm."

Let's start with question 2:


If you flip a switch to turn on a \$300\:\textrm{W}\$ xenon lamp, do you expect the switch to get hot? No, you don't. Likewise, if you can use a transistor that will guarantee a low voltage drop, even a small one can switch a large current. Power dissipation is about the device itself, as a switch. And the figure you give, \$1.4\:\textrm{W}\$, is a maximum and only in the case when ambient is \$25\:^\circ\$C and where you only pulse it so that \$t\le 10\:\textrm{s}\$. You need to be careful here, as they are using a thermal resistance of \$90\:\frac{^\circ \textrm{C}}{\textrm{W}}\$, which I believe was based on the maximum shown in the first line under "Thermal Characteristics."


Now for question 1:


See #2 above. That discusses power dissipation of the switch, itself. So you already have half your answer. The rest, switching power, is a term I can't find on the datasheet. I suspect you may have read it in a context where it's discussing the load being switched (as in that \$300\:\textrm{W}\$ xenon lamp I mentioned earlier.) Or else you saw it in the context of a CPU where there is "static power" and "switching power," with the switching power being a (nearly) linear function of operating frequency. If it is of this last kind, then it's going to be about operating this device very fast. And since you haven't talked about that part of things, I won't either.


Now:


I don't think it's worth nuancing over, given how little you've discussed about your future intentions using it. So, looking at the datasheet as well as what you've mentioned about possible \$V_{GS}\$ value ranges, I'd just say that you should plan on no better than \$50\:\textrm{m}\Omega\$ for it. You don't say anything more about your loads, so all I can offer from here is to suggest that you carefully watch those thermal characteristics for the device and check Figure 9 regarding the safe operating areas.


Just to expand a little on #3, where you want to know the maximum current you can use. This depends on a lot of things. If you live in Adelaide, Australia, where the ambient can be insanely high and if your device might be used inside of a car which might be sitting on the dashboard on a hot day there in broad daylight for hours at a time, the answer could be \$2.8\:\textrm{A}\$. Why? Because the inside of the car can actually reach almost \$100\:^\circ\$C. And the datasheet appears to say that under no circumstances should you allow the device's internal temperature go over \$150\:^\circ\$C:

$$\sqrt{\frac{150\:^\circ\textrm{C}-100\:^\circ\textrm{C}}{125\:\frac{^\circ \textrm{C}}{\textrm{W}}\cdot 50\:\textrm{m}\Omega}}\approx 2.8\:\textrm{A}$$

And that is pushing the maximums. So you actually should not want to do that.

On the other hand, if the device will be used pressed up against a block of dry ice in a situation where the ambient will likely be \$-50\:^\circ\$C, then you could probably push more current through it. You might be allowed to reach that maximum current you mentioned. At least you'd be in the neighborhood.

Again, it's pushing the thermal maximums. So again you actually should not want to do that, either.

You also may want a "margin of 2" on the current. Pushing the maximums while the inside of a hot car is kind of nuts. So is doing the same with dry ice. To figure something more normal as an operating temperature -- say \$85\:^\circ\$C -- then I'd say you should not plan on more than somewhere around \$1.5\:\textrm{A}\$ to perhaps \$2\:\textrm{A}\$ for the device.

Oh. And I should note here that I did not take into account anything that shows in Figure 9, above. This is where you need to know something about your \$V_{DS}\$ (which you didn't discuss) and the duration of your on-time (which you also did not discuss.) But you will need to consider the safe operating area, too.

We are moving into an area where opinions may vary somewhat. That's mine, sight unseen.

Looks like a nice device and it looks like Arrow is selling them dirt cheap, too.