Concrete question(s) about garage
The first question in regards to the channel outside the garage floor is to leave it be. It appears to me to be a sluice-way cut into the concrete to  direct rain water away from the interior floor or to channel it in a safe direction.   The photo doesn't show the bottom of the channel, but it's probably was cut after the driveway (not the garage slab) was poured (my assumption). If you are concerned about water undermining or eroding the concrete you can verify either by cleaning the channel out. If you see sharp angular corners the flowing water hasn't effected the composition or integrity of the slab.  If you prefer not to have an open trough crossing your driveway there are channel grates specifically designed to dress-up open drainage channels.  Lastly, to answer your last concern regarding the garage floor expansion joints; leave them be. They are purposely formed into the slab when the concrete is still un-cured in order to reduce stress cracks that may occur. The idea is if you engrave joints at regular measurements (and keep them relatively deep) any forces that may act on the concrete won't produce cracks in the slab face , but along the thin control joints.    If you paint the floor filling the joints shouldn't have any detrimental effect on their not working. It might be wise to not flood the joints with excess paint.