From working in the lawn care business for years and years:
No, you do not need 5 or 6 applications to your yard per year. They will insist that you do, and that this is the only ‘program’ they offer. Be firm. In the Mid-Atlantic region anyway, here’s what most lawns thrive on: an early spring fertilization with preventative weed control (keeps the crabgrass from coming up), a late spring fertilization with broadleaf weed control (for dandelions), and a fall fertilization. Have your local extension agency do a soil test for you to see if you need a lime or sulfur application.
If your lawn has a big problem with weeds, you need more grass. In the fall, aerate and seed your yard. Follow the watering instructions. Do this every fall until you have thick grass, and that will keep most of the weeds at bay.
Unless you live in the desert, do not plant Zoysia grass.
There is no such thing as ‘natural’ weed killer. Well, not one that works anyway. What you want is someone who will spot treat your weeds, rather than spraying weed control over the entire lawn if it doesn’t need it.
That stuff that looks like dog vomit in your mulch? It’s a fungus called… dog vomit. It thrives in dark, moist locations. Just shovel it off and throw it away.
Water your lawn/plants in the early morning, before it gets too hot. Watering in the middle of the day can burn the grass (sun + water droplets = magnification = burns), and watering at night promotes fungus growth. Also, water deeply.
Set your mower blade on it’s highest setting, at least 3 to 3½ inches off the ground. Try to never mow off more than ⅓ of the blade length at one time. Keep the blade sharp. If your grass gets a whitish/grayish fringe at the tips, your blade is dull. Leave the clippings on the grass… they are full of nutrients.
“Pesticide” is a very broad term that includes things like Clorox bleach, Lysol, Purell hand sanitizer, and mouse traps.
IPM (which @cprevite mentioned up there ^) is a good thing to look for in a lawn company, too. It stands for Integrated Pest Management.
Industries like this one that use a lot of water can get special permits and a tool that allows them to take water directly from fire hydrants.