My husband and I recently bought a house that was a foreclosure for 1 1/2 years, hence a yard that needs a lot of work to bring it back to health. Both of us are newbie gardeners and would like to take care of our lawn organically. The soil is hard and compacted and in need of some life breathing into it! We are going to get that tested, but meanwhile about mowing.. we've been bagging all the clippings but read it's better to just leave them on the ground as a kind of mulch. Thing is we have dandelions, bindweed, thistle (a few different kinds), loads of mallow and a number of others not identified yet. I understand that the flowers can have seeds and so would make sense not to leave those lying around after mowing, but what about the other parts?
I've read that cutting bindweed at the soil level is best and not letting it go to flower, not sure what do do with the mallow though, we have a lot of big well established plants with flowers. I understand the mallow roots run very deep and if we were to attempt to dig it up we'd have huge holes in the lawn, shall we just mow over the mallow for now? The thistle seems to shrivel really effectively with a vinegar/soap solution sprayed onto them.
"The soil is hard and compacted" - Mow it and leave the clippings. Put the organics back, and this will help. If weeds do sprout from this, they're shooting roots in and helping the hard and compacted soil, helping the compaction. And you'll mow them again and both feed more organics and weaken them. Weeds are green, and your soil wants green.
You might be able to play with the height some, depending on if/what grass exists, and if you want to keep it. "Generally speaking" I'd still mow to a height that isn't scalping (so you have some green to look at) until the fall when lawn renovation can take main stage (assuming cool season grasses).
While you're figuring things out, simplify and feed the soil: Mow, leaving the clippings. Don't make more work for yourself with bagging. (Unless you really want to, and then I'd suggest packaging it up and sending it all to me for my compost pile! )