Which resistors are fitting for the mosfet gate resistor?
10 ohms if you are going to do any PWM dimming, but 100 to 470 are fine if just turning on/off once every few seconds to hours.
I still have no idea what the resistors after the LEDs are for and how much resistance they got to have.
Post the LED or light strip you plan to use.
Should i pick another board instead of the ESP8266 since i wont use WiFi at all? Would e.g. the digispark pro?
It depends on what you are trying to do with the light strips (dimming, signaling, etc. the esp8266 will definitely work - it is very fast chip - 80MHz. Remember, the I/o pins only output 3.3v so you need mosfets that turn on completely with only a 3.3v signal.
In summary, let us know what you are doing so we can answer.
No, you should use a 3.3v regulator. If the current draw is under 100mA, you can use the regulator that sends 3.3v out to the "3V" pin on the ESP8266 (if it has one).Should i use a voltage divider to get the desired 3.3V at the BT-Module input?
Yes, install as shown in datasheet with caps.7805 Voltage Regulator is specified with a max Input Voltage of 25V, will that be a problem? Do I need additional parts like capacitors for this voltage regulator?
Why do you use 28v?