I think this is doable, and (surprisingly) the laser is well documented.
What I don't see is anything about a movable mirror or anything to control "engraving"... Is that part of this project?
Im just beginner at Arduino
Do you know any electronics? Do you do any programming?
I'm physicist...
So, you understand something about electronics... Do you remember any Boolean logic from school?
The Arduino is not actually TTL, but it is TTL compatible. That means if you write a LOW (logic 0) out of the Arduino it will put-out (nominally) zero volts and if you write-out a HIGH (logic 1) it will put-out (nominally) +5V. And if you connect those outputs to your laser inputs the laser will read the states correctly. Similarly, the Arduino can read the laser outputs (alarm condition) on it's inputs.
I didn't actually count the number of connections required but the regular I/0 pins on the Arduino can be programmed for either input or output, and there are pins that can also be used for analog input and pins for PWM output. I think you have enough I/O pins, but check that for yourself!
The only thing that looks "tricky" is the Pulse Repeating Rate. The Arduino has tone() function that puts-out square waves. It runs at a lower level on the processor so you can just set the frequency and start and stop it, and it runs independently of your other code so that should work for you.
If it weren't for the (dangerous) laser I'd say this is a good "beginner project", and since you are experienced with lasers it's probably a good beginner project for you. The ONLY thing I'd worry about is somehow damaging the laser but sending "invalid" commands/states into the laser, or your code not handling the alarm conditions properly.
During development, you can "simulate" the laser and test the Arduino & code by connecting LEDs to the Arduino outputs and switches/buttons to the inputs. It might be a good idea to leave the LEDs in your final project so you can easily see the output states, and you could also put LED on the laser's digital outputs (alarm outputs, etc.). The Blink Example shows you how to connect an LED, and the Digital Read Serial Example shows you how to connect & read a switch.
BTW - The serial monitor (see Digital Read Serial) can be very handy for testing/debugging... You can send all kinds of helpful information/messages//states/variable-values out the computer so you can "see" what the Arduino is doing.
