In 4 layer stackup signal/ground/power/signal, can we route on layer 3 without transfer vias as there is no change in reference plane (layer 2)?
Sure, as long as you are not designing very critical systems (such as DDR3, DDR4 or those multi gigabit serial signals (USB3, PCIexpress etc) then having a single continuous GND plane is usually sufficient. For the GND plane there should be no big discontinuities. If there a lot of via's in some area, then make small groups of them, so the GND plane connects to itself in between the via's. The other thing that should be abundantly clear from Robert Feranec's video is that the return current follows the path of least impedance, (above a few kHz signal content) and not the path of lowest DC resistance. For any signals above a few kHz, the impedance is determined by the enclosed area. Big enclosed area's work as loop antenna's. Just look at the pictures from the link below:
https://duckduckgo.com/?hps=1&q=loop+antenna&iax=images&ia=imagesThose loop antenna's are good at radiating energy and at catching external electromagnetic fields, and this is exactly the opposite you want on a PCB. To minimize the loop area, think about the return current of each signal, and what is the closes path though the GND layer for each signal. Also think of this during the fan-out from IC's.
As for the other layers. You can pretty much change the routing for both power and signals on whatever layer fits best. On an ESP8266, the most critical you will encounter is probably a quad SPI channel to a uSD card. It's probably best to keep this routing on the signal layer that is closest to GND. Routing around crystals is also a bit of a special case.
And in the end, it depends a lot of what your intentions are. If you want to make a few DIY PCB's for home projects, they will probably still work if make an atrocious PCB. Hand wired Matrix board has been used for many years and those don't have a GND plane at all. (But I have noticed they are quite sensitive for picking up interference from for example switching the ballasts of nearby Fluorescent ligtning, or the transformers from halogen lights).
If you are designing commercial products and have to verify and test your PCB's for EMC compliance, it's a whole different story. It's still the same rules, but you have to apply the rules far more rigorously.
Also, any connection to the outside world (including the power wires!) need special attention to prevent them from working as antenna's, but that is another part of PCB design and a whole chapter in itself.