Re: the toe toe. No, not common and I wouldn't expect it to be. Considered an invasive risk 9 plant by APHIS, although a great many plants native to the southern hemisphere and Australia/New Zealand in particular are not considered reliably winter hardy in this climate, only borderline at best. And that includes phormiums, most eucalypts, hebe, acacia, grevillea, pittosporum, coprosma, drimys, dianella, etc. They may live 3 out of 5 winters but that 4th or 5th will be the end of them. The exceptions are the hair sedges and the wind grass (anemanthele), which are very hardy, sometimes to the point of weediness.

I suggest spending some time studying the Ornamental Grass Comparison Chart (link below) listed by Bluestem.ca They specialize in grasses for cold climates, and their website has tons of good information and photographs. Also look at their Landscape Uses section, to find lists of grasses for specific landscape conditions. Here is a link that might be useful: Ornamental Grass Comparison Chart

Where do you live? What kind of grass do you have? - Dethatch No. Thatch, if you have it, is the least of your problems. - Cut the grass short Oh heck no. Always mow at the mower's highest setting or nearby. - Scarify Scarify is something you do to seeds that need to be roughened up. What does it mean in your context? - Aerate Probably not. Wait until you have the major issues resolved. - for the muddy areas, apply triple mix soil to flatten and level the lawn Not yet. Assuming you have low spots, you need to address those. Fixing those should get rid of the muddy parts. - overseed / over the mud spots, I will apply more seed and Wait for seeding (preferably until September), but until you have the bumps and dips sorted out. - put triple mix soil on top of seed No. You probably already have too much soil in the yard. Almost nobody is short of soil. There are tons of people who put 1/4-inch of topsoil on the yard EVERY year. Well, in 4 years that's an inch of soil you don't need. - Rake and spread the soil around NO! No more soil. - add water NO! You already have a water problem. I live in Texas and have not watered my lawn yet this year. Well, a little on the southern bank, but then it rained, of course. The big thing is to identify any drainage issues. Water should drain away from all the buildings and out to the curb or a ditch somewhere. If you have low spots anywhere, or if you have soil higher than the concrete walkways or driveways, then you have too much soil already. Scraping it up and hauling it away is a huge deal that I've rarely seen done, but that solves so many issues. Fixing the drainage will NEVER involve tilling the soil. It might involve scraping surface soil away to form a swale or ditch, but rototilling is never in the solution to drainage. Rolling is similarly useless in profiling the surface. It looks like your yard is bumpy mostly from lumps of grass interspersed with dead spots. I would not jump to any conclusions about the soil being bumpy until you get an even stand of grass.

Because commonnand hybrid Bermudas share a name, people tend to think that they're basically the same. While a dissected blade is going to be very similar, the overall plant habit is very different. Hybrid is dense and low; Common grows tall and loose. Personally, as I like things tidy. I would not want the latter unless we were talking about for 40 acres on the farm and were going to mow it with the tractor. Then fine.