Layout design in a board with Ethernet Controller and Connector
1) Keep 2 layers or go through 4 layers? (1 gnd plane and 1 power
  plane, 2 signal layers)


If the board is already routed, then keep 4 layers. I would think it would be difficult to keep the impedance controlled lines with a continuous ground with 2 layers.


  2) How to make a correct differential pair for Ethernet controller
  wiznet W5100, I did some serpentine to tune impedance on 50 ohms, and
  tune differential pair skew to 0mm.


The width of the traces, and the height above the reference plane and the relative electric permeability of the material in between is what constitutes a differential pair. It's best if the  diff pair's reference plane is continuous below the differential pair. Ethernet differential pair impedance are 100Ω, use a calculator and find the trace size for your board.


  3) Theres a need to use serpentine together with differential pair and
  which rules I should use, general rules, i find this reference design:
  https://wizwiki.net/wiki/doku.php?id...hardware:start
  
  5) I have SIM800L module, Esp8266 uController and Ethernet interface
  what to improve looking for this design and its reasonable this way or
  need to make rework, like place multilayers and its correct to use
  vias to do differential pair for ETH as was done? Well, whole board
  was done using auto router by coworker here...


A good place to start would be to keep all the differential pairs on the top layer, and not cross them. If this is your first time routing differential pairs, you probably don't want to find that the board doesn't work well and you have to build a new one. If you don't have a way to measure fast signals (like a +300MHz scope with differential probes or a network analyzer), if something is wrong, you won't be able to measure what is wrong. 

If this is the case then you should be very careful in routing the differential pairs, they should take first priority. The additional cost of a 4-layer design would be minimal to the time it would take to debug and re-spin new boards.