71 Duster carpet 4 speed

Do I cut out the shifter opening but it lays on the lip where the boot and trim screw into?
Does the boot screw down through the carpet to the tunnel and then the trim ring over top also to the tunnel?
Yes and yes, but if the shifter is installed, it's a lot more difficult, cuz you're not exactly sure where the hole should go. and if you put it in the wrong place, well, that about screws it up.
Honestly, I would drop the shifter.

Here's how I installed mine, and it went pretty smoothly.
I threw the carpet in there, centered it, and made sure it was laying where it was supposed to, with the dimmer-grommet on the dimmer-switch. (my grommet was pre-installed.)
Then I trimmed the outer edges a lil long, pushed them inboard to allow for climbing up the lips, then installed the sill plates as clamps.
Then I cut a hole for the shifter, about a half inch smaller than the hole in the tunnel, so I could find the edges of it, then installed the TRIM-RING on the metal tunnel under the carpet, then trimmed the carpet right out, using the trim-ring as a guide.
Then I removed the trim ring, and installed the boot, onto the metal tunnel, using some awls down thru the trim-ring holes to locate it in the correct place. With the screw holes thus established, I took the boot off.
Then I re-installed the shifter,
followed by the boot right onto the metal tunnel with it's short dull screws.
In my case, the trans was not yet in, so; from the bottom, I punched my Awl up thru the trim-ring holes in the tunnel, and thru the carpet.
Then I installed the trim ring on top of the carpet, with it's shiny plated screws, down thru the just pierced holes. Do not overtighten these as the trim ring will buckle.
And finally, I installed the ball and locked it down.
Now, if your trans is already in, you'll have to locate the Trim-ring holes from the top. So yur gonna need some really slender pointy tools to go hunt for them.

At the seat mounting holes, I took my Olfa knife, with a new blade, and cut an X over the holes, then trimmed the tips of the pointy bits off with a tin-snips, to allow the seat studs to pass thru, for a pucker-free install.

That's how I did it. In hindsight, I guess I could have drilled ALL the holes before the carpet went in, but I didn't think of that at the time.

Here's a couple of tips, for if you have an A833;
1) In the rain, water does NOT come up to that hole. But
Dust and gravel sure do. So then,
if you travel a lot of gravel roads, like I do, my shifter accumulates dust, and fine stones; so I never grease/oil the shifter guts which would only make it wear out faster. Every once in a while, like once a year or so, I remove the boot and blow the shifter out with compressed air. My shifter is circa 1971.
2) the external shift lever nuts, and the adapter plate which the shifter bolts to, they tend to work loose, and then shifting gets sloppy.
To combat that, I red loc-tite the cavities around the levers before the nuts go on, and I blue loc-tite the adapter retaining bolts then torque the chit out of them. The big bolts that mount the shifter to the adapter, for me have never made trouble, not in 50 years, so I put nothing on them.
3) I highly recommend to never use synthetic transmission fluid. That fluid is extremely slippery, and I can almost certainly guarantee that your brass rings will have trouble synchronizing, which will make your shifter seem slow. In my case, unacceptably slow. I mean I really really tried to make it work, taking my trans down four times in about two weeks, to try different things. In the end, I took it down, and apart one more time and washed all that synthetic out.
Then I installed 100% Dextron II and the doggone box shifted like lightning! All the mods I did, now worked. I now shift at 7000/7200, and never miss a shift.
A year later, I noticed some wear on the cluster-pin, so changed the oil recipe to 50% ATF, 50% 85-90 gear oil. Still shifts about the same, so that's where it has stayed since 2005 or thereabouts.
4) the factory neutral/reverse switches like to leak right thru the black potting. Even new ones. So, before I install them, I clean them several times with Brakleen and compressed air, then seal the perimeter and the electrical lugs with either red loc-tite or epoxy. Epoxy has worked well for me.
5) BTW, I have been running the A833s almost continuously since 1970. In the beginning I had a pretty low opinion of them cuz I kept breaking them. But after learning why I was breaking them, and the proper work-arounds, I soon got over that opinion. Now I just run the heck out of them. I haven't broke anything in this latest one, not in well over 10years.
But, I gotta admit,
Second gear does need a new brass, again, as she's shifting a lil slow, lol.