You should not use a grass catcher. The decomposing grass pieces provide natural nitrogen to the soil.

It is more likely the grass is Kentucky bluegrass (an excellent turf) which they watered daily (a poor practice). That will result in thatch buildup which looks like what you're describing.

I would not rake it up just yet. I would get in the habit of watering deeply and infrequently, follow that through the summer, and see if you have any thatch left in the fall. Then deal with it. Watering deeply means one inch all at one time. Measure one inch by setting out empty cat food or tuna cans and time how long it takes your sprinkler to fill the cans. It could be anywhere from 20 minutes to 8 hours depending on the water pressure and type of sprinkler. Once you know how long it takes to get an inch, then water for that length of time when you water. Infrequently means once per month in the cool part of the year (now) and moving to once per week in the heat of summer.

For grass always use a dry type fertilizer. I like organic all the time, but even if you insist on using chemical, I would urge you to use one application per year of organic. You can do that any time of year even if you just used the chemical fert.