If that's the going rate, that's the going rate. Weekly is easier for them, for you, and for billing purposes. It is probably also good for the grass as you don't want to take off more than 1/3 of the leaf at any time.
Think hard about your intended backyard use. Will you be having picnics, playing ball, putting down sprinklers, using lawn toys, playing, croquet, a garden, making a fire pit, installing a swing set and fort etc? Think about the footprint of grass or other walkable surface you need for that activity.
Allow yourself a few extra feet, then you can either let the rest "revert" to natural, or you can purpose plant with alternatives like - Meadow/ wildflower seed. Perrenials, Shrubs, low lying ground covers, trees.
Call your local politician and ask to be put in touch with someone from the agg extension, they'll do soil testing and let you know what you are fit to grow and if your garden soil is safe for food production.
Also, observe, observe, observe. Table the project for a year, draw a map of your yard, overlay with quadrants and walk the property at least weekly making observations and jotting them down. Do certain areas get a lot of sun, dappled shade, afternoon sun, are some prone to pooling, do some get muddier than others, are some soft underfoot, does ice set in in certain places etc. With careful notes, you'll have a better understanding of your own particular "micro-climate" and can buy plants accordingly. Buy the "wrong plant" for the "wrong spot" and you'll either loose it, or once again long for the days when it was all lawn and you were paying $400 a month to cut it.
As for the composting, its as easy as pie. Decomposition happens naturally, jut pile up your lawn waste, leaves, branches. It will turn to compost on its own, with enough time, even if you do nothing at all to it.
keep it moist not wet, turn it with a rake to get some air working through it, and it will compost faster. If you are concerned about inviting vermin, don't add any food waste, and buy or make something out of a metal drum.
Ratios of browns/greens what to include what not to include, how to know if its done, how much to apply and when are available on the internet, at botanical gardens, and through your county agg extension.
Good luck to you and yours,
F. B.