Strategy for a learning lawnmower robot, and choices of motors
I don't think it will work in that simple form.  There will be variations of the turns and distances despite your best efforts at recording them.  Even on a flat, smooth surface you will have problems with repeating a particular course perfectly.  Add in bumps, dips, and varying rolling resistance and I really doubt your robot could follow an arbitrary course perfectly every time.

I would suggest trying it out with a simple robot first.
There are many simple (and cheap) wheeled robots available.  Just program one to follow a set course (forward so far, turn right so many degrees, forward so far, turn ,etc.,) then run it several times to see how consistently it can repeat it and how far off course it can get.  If it does tolerably well on a smooth floor, start adding bumps or other small variations and see what happens.

What you are proposing is based on dead reckoning, and depends on very precise control of distance and turning angles.  Any variations will compound with time and distance traveled, until at some point you are completely lost.



If you equip your robot with a navigation system, and record the commands as waypoints, then you might get some where.  It knows where it belongs and where it is allowed to mow, and does its best to stay there and follow the given track.
You would, however, have to have some kind of way for the robot to know where it is.  A GPS receiver is a start, but it isn't precise enough.  Maybe differential GPS, with a separate receiver as a reference and bluetooth/wifi to connect with your robot.
Inertial tracking might help. Costs are going up, however.
A smartphone would have all of the bits needed for tracking and navigating and would be cheap but is has %$&amp;$% all in the way of outputs - you might misuse the earphone or use bluetooth for the phone to send commands to the hardware - and just as little for inputs - again, bluetooth is a possiblity but its bandwidth isn't unlimited.

Building your own controller means interfacing with all the needed sensors and navigation tools, and so might be more expensive.

All in all, you've picked yourself a fun project with lots to learn.
Motor control, battery management, programming, control systems, feed back loops, systems integration, safety considerations, navigation and lions and tigers and bears, oh my!