First, the signals
It doesn't make much sense to try to segregate the signals by assigning them layers depending on their kind. The fact you have two layers for the signals allows you to make crossings. If you have a lot of digital signals, for example, you need two layers to route them efficiently. So, use both signal layers for all kind of signals.
What makes sense, however, is to separate the circuit in blocks of different type (high power, digital, analog, ...) and put them on different areas of the PCB, to minimize intererences between sensitive blocks (e.g. put noise-sensitive analog signals far from switching regulators, ...). Also choose the locations of all these blocks so that you minimize the trace lengths between blocks.
Now, the power planes
Whether you use the same layer for 5V and 12V (with a split plane), or use one of the signal layer to have them on different layers is not that important. There is one big rule to follow, however: avoid signals that cross a split plane (especially high-speed signals): that would create unwanted EMI.
Check what makes sense in your specific case: if the 12V is used only in circuit parts that do not use the 5V, use a split plane. If the 12V is used only on a few nodes, but in circuit parts that also use 5V extensively, use the whole power plane for the 5V and route the additional required 12V traces on a signal layer (you don't even need to use a plane, actually).
The only thing is that, if you use the whole power layer as a 5V plane and have a large portion of the signal layer on top of it with an additional 12V plane, there will be capacitance between them. So if there is noise on one of the supplies, it may couple on the other one. So maybe avoid that.
My feeling
Your design doesn't seem to be very sensitive. I don't see very high-speed signals, you didn't mention switching regulators, ... So actually, anything will do. Just follow you gut feelings and logic to make choices (minimize trace lengths, organize blocks efficiently, ...), and it will be fine. And probably Roger Rowland is right in his comment: two layers would most certainly be enough. You suggested that 4 layers allows for "better grounding", but there is no point in making the ground "better" (whatever that means) than required, especially if it costs 4 times the price. Careful layout with gridding the planes on the two layers would for sure provide a good enough ground in you case.
Here is a document from TI giving good advices (and in particular the ground gridding explanations in chapter 2.2.3).
Manufacturing a board, from the customers point of view, is no different no matter what you draw, unless you add excessive numbers of holes to the board. It is much more involved to make a 4 or 6 layer board than a 2 layer board, and the cost and time will be greater. Multilayer boards allow a ground plane and power planes to be used. Once you settle the manufacturing (number of layers, layer stackup, minimum space and trace width, maximum hole density, 'via' technology and minimum size, minimum annular ring, etc.) the cost will not vary much.
Assuming you have a two layer PCB you don't really have the option in most cases of a complete ground plane, because otherwise you would have to lay out your circuit as a single layer (excepting only ground). So your options are pouring or not pouring.
If all or most of your parts are on the 'top', you can often pour a ground on the bottom that is mostly integral. If you care about EMI it's better not to have high speed signals crossing a break in the ground pour or ground plane (you can split planes). You may also choose to pour on the top (where the parts are). In circuits where there is mostly one ground and one supply, it may make sense to pour a ground on the bottom and a supply on top. The benefits of the latter in particular are not so great so you may want to make sure you leave a generous clearance so the yield is not unduly adversely affected. In other words, if the PCB maker says they can do 6 mil clearance, use 15 or 20 mils for the pour clearance, not 6 mils.
The distinction between a 'plane' and a 'pour' on a multilayer (4 or more layers) board is partly the way they are drawn- a plane is drawn in the negative and you may split it (for example to provide a second ground for galvanically isolated parts) whereas a pour is put overtop of conventionally (positive) drawn traces and pads and connected to a net. Either can provide connectivity, so you can eliminate any traces that were there providing connectivity. If you neglect removing those traces you can muck up the thermal reliefs a bit but it should still work.
Either eliminate dead copper (unconnected islands) in the pours or stitch it to connected sections with vias and short traces. In this way you can get a mostly integral ground layer and improve the power distribution at no additional cost.
Best Answer
Here are some general pointers since all we know at this point is that you are driving some high currents (12A) from a battery pack:
The ground and power planes are a good idea when there are going to be many chips / devices connected between them, distributed across the board. The planes set up a sort of distributed capacitance and low inductance which is effective at high frequencies. It is not a good idea to have the power for your motors flowing through the plane. The width of the trace and copper thickness determine the conductivity and total copper loss in the trace: outer layers can dissipate more due to convection.
Identify your high current paths. This will typically be from the battery +ve terminal to a MOSFET/switch, to the motor, and back. You should try to keep this path length as short as possible: perhaps even using wires instead of PCB traces.
You also mention that you
need to draw only one 0.5mm trace line for 1.8v from layer 3
which I suspect is your control line. In this case you should lay it very carefully to ensure that the motor currents do not cause ground bounce.In summary, it would be better if you were to post your circuit (some abstraction should be ok) in addition to how you plan to lay it out; and ask for feedback.