This is interesting! I took a magnifying glass to see if I could see thatch, grin! Thatch would make sense. Look on the edge by your sidewalk, you should be able to see a 'mat' of roots, un-decomposed lawn debris and very little soil. If it is more than an inch thick, it is time to de-thatch. I am not familiar with your type of grasses but your observations about your neighbor's influence and the difference in vitality between your grass and theirs is excellent. Seriously. I'm a Landscape Architect I thought I'd jump in and opine.
Is your home the brick home? Just half of it or all of it? Why the chain-link fence? Is there more to your lot where you stood to take the pictures? Love to see your front yard as well. Do you have ANY plants other than grass? Love to see your home, roof and all...
I'm just going to come out and tell you my thoughts and if I am wrong, just let me know.
A small yard like yours could be easily designed to be an extension of your home with outdoor 'rooms'...privacy, simple foundation planting, a small, simple water feature for sound or white noise, beautiful screen panels made with 2X2's, lap-jointed with 8" thick posts staggered (versus a 'fence') allows the eye to stop at the screen defining your outdoor rooms while allowing air and neighbors to feel welcome. An arbor or small tree canopy, a bright and cheerful awning or a simple porch (and back door) would allow you to sit outside during a storm sipping on hot chocolate. Ha ha, how about a fire pit? A hot tub? A simple patio with groupings of potted plants?
A professionally designed landscape would vastly increase the value of your home. Far more than remodeling the kitchen or adding another bathroom...are you going to live in this home for the next 5 years, 10 years...or sell next year? I see such possibility to improve your living experience, investing $4-5,000, easily increasing the value of your home by $40-50,000.
What I see in your pictures is wasted space. Maintaining anything that you never enjoy is a waste, my opinion...are you interested in enlarging the living area of your very own home? Do you have friends over? Are you married? Any pets? Kids someday? Would you like to grow your own food? Do you even like being outside during a storm? Do you like to cook outdoors? Eat outdoors? Do you find you are disconnected from 'nature'? Grin. You would be amazed to see what I envision after seeing these few snapshots.
I hope that you'll reconsider growing just grass (gees! Get a sod cutter, cut it all out, use it to make ornamental plant beds, put in a vegey garden...replace it with new sod, fertilize with an extended release organic fertilizer-some have bacteria that eat thatch), aerate once a year, water deeply and allow the soil to dry out to get your lawn to dig deep for water and become drought tolerant or...
Begin thinking of your home and landscape differently...small can be big!