No. But you will still need a logic level FET to do the switching.
But you do need to supply more information about the strip. Are they addressable? What is the current each does each LED take?
From the information supplied so far it looks like you will need a power supply of 12V at least 2.2A if not a bit more.
Let's say I power led strip with over 12V 5A adapter, and Arduino with 5V adapter, will I be able to control logic of photoresistor (for my project mentioned above - light/dark switch off/on) via Arduino or do I need something else?
Yes that will work.
You will need a resistor on the photo detector about the same value as the mid photo resistor’s value going from the analogue input an 5V. Then the photo resistor from the analogue input to ground.
Also you will need two resistors on the FET. One of about 200 to 500R in seriese between the gate and the Arduino output and another one of 10K value from the Arduino output to ground.
No you can not power the LED strip through the Vin on the Arduino. That pin is connected to the power jack through a 1A rated diode, and as we have established that you need more than this then you will burn out your diode if you try this.
You also have not put the resistors in the FET's gate like we discussed earlier.