Okay a few things.  

1-No, don't bury the sod.   But you DO need to hollow out a little space, and "set" the dirt part in the ground, and make sure the edges have dirt sloping up to them.  Basically, when you have a big piece of sod, you have the middle that does not dry out completely, and the edges that get very dry, if they are not butted up against something (which they almost always are).  With small pieces of sod that you are treating like plugs, you've got to sort of "plant" that piece of sod. Use the soil to help it not die from drying out too fast.   Even with you watering plenty, a small piece, without a root ball (like a plug would have) is going to get very dry, very quickly.  So this is more work than just rolling out strips of sod, but it will pay off for you.


2-Your sprayer is fine.  Herbicide for the weeds in zoysia is what you want. So you want a herbicide that won't hurt the zoysia/warm season grass. It will be primarily 2, 4-D, but usually the formulations have something else in there too, to give it more punch.  They are usually generic and very cheap in small bottles.  You are going to overspray your turf, and also spray your bare areas.  Folks seem to really like Celsius, but you should research that.

3-I tell my customers to go to Lowe's or Home Depot and get decent quality impact sprinklers from there.  Cuz they are not gonna order something specific. They just want to buy it, shove it out there, and have it work.   The ones @TheStig posted are absolutely fine.  I could recommend some truly outstanding sprinklers, but they are more expensive, and you don't need fancy just for establishment.  I do have some opinions on timers that I absolutely hate and you should not buy.  Pay the money and get the one you can run from your phone.  I can't think of the name of the rotten one. If you need that, I'll go look at the POS one I tossed in the garage.


HERE IS THE TROUBLE:

Zoysia is slow to establish. It does not spread quickly like Bermuda does. Especially at first. It can take a couple of years to really get established and covering ground.  That means you are going to be spraying that bare ground a LOT, and you are likely to go over your annual rate for the herbicide use.   This is more than just a problem on paper.  1- Weeds develop resistance, just like bugs do.  Overuse of one herbicide can make it less viable for you in the future.  2- While we say "X herbicide doesn't hurt Y grass" that's not entirely true.  The grass still has to process that herbicide, and it's going to slow down the establishment and spread even more.  

So you've got yourself kind of between a rock and a hard place here.  I do NOT recommend using glyphosate "between the plugs of sod" because you're accidentally going to kill some of it.   You can't use a pre-emergent herbicide like prodiamine because that's a root pruner.  It will keep your sod from rooting, tacking down runners, etc.  

Honestly if you can swing it, I'd either sod the whole thing, or  go with a different, less-expensive cultivar and just redo the whole thing.  Maybe put the fancy sod on a curb strip or something so it all matches in one section?

I dunno.  This is a tough one.  I wish I had a better solution.

Yes, ask questions, and show us how it goes!

So if you are going to aim for, say,  a hundred square feet per month. (just guessing...it could be half that...no idea what you are thinking) MAKE A PLAN.  Grid off your yard, and make a paper plan.  DECIDE NOW which are the first three squares (300 square feet) you are going to sod.

THEN...put prodiamine down (this is a pre-emergent herbicide) on EVERYTHING ELSE.

This will save you so, so, SO much hassle.  

I would also consider getting some bags of peat moss at Lowe's or HD, and spreading that over the bare soil, just to help it hold water and not blow away, turn into a complete mud pit, etc.  But That's a "nicety".  What you really need is weed prevention.  The pre-emergent is your friend, AND...if you do the application correctly, it will wear off in....drum roll please.....three months.  Just when you are ready to start your next square of sod.

OR you could get landscape fabric and a bunch of staples, put down the pre-emergent, then roll out the fabric and staple the hell out of it.  (And lay down some rocks, pavers, etc.)   It won't be beautiful but it will help you in the long run.  (Or you could do black plastic and no pre-emergent.)

Yes  If you haven't, you need to get a little fertilizer on that sod.  Something with phosphorus in it is good.  That helps with root development.  Like what I use on new sod is, I put down a granular starter fertilizer on the bare dirt before I lay the sod (this is NOT a heavy application.  It's just a little.)  Then I water it like crazy, then I apply RGS, (this is a liquid) which is Greene County Fertilizer's root growth stimulant.   Water that in.   Then in about four-seven days (depending) I apply a liquid 16-21-2 with humic and fulvic acid.  

Now, I grow cool season grass (fescue, bluegrass, etc) but I don't think this sod starting technique varies much between cool and warm seasons.

Very nice domination line!

How many thousand square feet? (I'm wondering how much that 100 lbs was, as far as pounds on the ground per K).

We want pics, dude!  

You got any Greene County Air-8?

Throw that to it, as SOON as you core aerate.

It is HUGELY helpful to get it in the bottom of those core holes!

That's called Cleavers.  I'm making medicine out of it today.

I expect 2,4-D will get it, but I've never tried to kill it.  I will look.

ETA: Oh...it's in a flower bed.

Shoot.

I don't know anything you can use to overspray ornamentals that will take out Cleavers.

Look for the little spiny balls (they're really small) on the plants.  It is a spring annual so it flowers and seeds very quickly, then dies with the heat.  But it leaves the  brown, ugly vines all over everything.  

Pre-emergent in the fall and early spring are your best bets to stop it. But that won't be effective THIS year.  That'll be for next year.  

Going back to read the comments.  You may already have gotten help.

@SWIRE

You're going to have to pull it by hand this year.   As you clear an area, sprinkle Preen all over your flower beds. (Try not to put it ON the ornamental plants, except for the vinca.   You can't kill that so don't worry.  Sprinkle away.)   You will need to reapply Preen every three months.

Toss the cleavers in a pile, then pick it up and put it in the trash.  It smooshes down much smaller than it seems like it would. It kind of collapses.  

FYI, this is not in fact goosegrass.   That's something else, and is a good reason to use scientific names, cuz regionally the names really vary.  But those scientific names can be really offputting.  If you type in "Cleavers" you will normally get this plant, though.  It grows pretty much everywhere around the country as far as I know, though maybe not in the deep south or southwest--not sure about that.  If @TheStig doesn't know it as a weed, maybe it doesn't grow down there.  


Since you have mixed perennial beds, you should absolutely read the label  HERE and avoid anything it says you should avoid.

It is my experience that NO granular pre-emergent herbicide is labeled for use around hydrangea (at least, it wasn't last time I looked) so I just stay back a little, away from the roots.

Pre-emergent is the key to low-maintenance perennial beds.

In GRASS, you can use 2,4-D and/or other broadleaf herbicides.  But not in your landscape beds.  

Unfortunately there is no selective herbicide that will get just the plants you want gone.  We all want that, but it doesn't exist.

So the seeds that get left from this year (and some from previous years, cuz that's how seeds roll, yaknow?) will lie there until winter.  Then they will germinate either in late fall or (more likely) early spring.  So it's important, in the fall, to put down a late fall application, and another in, say, late January/early February here.  You have to catch them BEFORE they germinate.

FYI I usually see Ginormous containers of PREEN at Sam's this time of year.  Much cheaper than going to Lowe's or HD.

You do also have chickweed in there, and of course a dandelion.  Tis the season.

Pre-emergent won't stop a second-year dandelion, but will stop the seeds of new ones.

I think your mom is lucky to have her own personal greenskeeper.

Yup, I kill it for my clients.  But I use it at my house.