I do have an old Mac, though it's in a fearful state at the moment, and it's so old as to be, likely as not, of little use in this survey.
All I know is my Mac is a late 2008 model iMac with a 24" screen, bought in February 2009 and which served me well until the end of last year when it suddenly and inexplicably kept on restarting with no warning. In May this year I replaced it with a five-year-old business PC that was given so many upgrades it might be considered (in xisumavoid's words, wasn't it?) a "hadron collider".
1. Can you run Minecraft 1.13 on your Mac? (i.e. no crashing upon launch)
No. In fact, I never managed to get a single 1.13 snapshot to run on the Mac - every single one of them crashed. I haven't tested the final release of 1.13 on it, because there's no point. Also, the launcher version 2.0 never, ever worked, and I was stuck on the old one that kept nagging me to upgrade with the message at the top. Is this launcher needed for 1.13?
2. Your current macOS Version
If I remember rightly (because the Mac doesn't always fire up these days), it was 1.8.5. I was specifically told by the people in the Apple shop that this was the practical maximum for the model I've got - 1.9 onwards would slow it down progressively further until the latest version would make it run at the pace of an early 1980s Apple IIe. I'm fairly sure this is the root of the problems with the launcher - I've seen it implied (though not outright stated) that OS X 10.9 is required to run the version 2.0 launcher.
3. Your Model Identifier
"2008 model iMac, 24" 1920x1200 screen, Intel Core Duo 3.06 GHz processor, 1 TB HD, 4 GB DDR2 RAM" is about the best I can do here.
4. Your GPU Chipset Model
I have some Minecraft screenshots from the Mac days which all say: "NVIDIA GeForce 8800 GS OpenGL Engine 2.1 NVIDIA-8.16.82 310.40.00.20f04".
5. The latest version of macOS available to you, if you haven't upgraded to the latest version yet
See question 2.
6. If you aren't able to run Minecraft: Does upgrading your Mac to the latest macOS version fix it?
It might, but there'll be no point in trying because it'll take far too long to even register I've done anything before it inevitably restarts again.
Whatever happens, there's no going back to Minecraft on the Mac, now that I've paid a grand for the new(ish) PC. It was getting silly throughout the 1.11 and 1.12 cycles - I had to upgrade the OS from 10.6 to 10.8 just to be able to run the version of java needed for 1.12, and it was slow enough that the memory would be fully allocated within a few minutes and I'd usually have to drop the render distance to 8 chunks or so just to be able to keep it going (and it was hardly ever at more than 11). Combat was tricky (especially with the 1.9 revisions) because I couldn't time my shots and would usually get quite badly beat up - only a maxed-out set of diamond armour (all with Protection IV) would save me, and a new world where I had to revert to iron armour was one hell of a wake-up call.
The PC is a 3.4 GHz 4th gen i5 (from about 2013), with the OS and Minecraft hosted on a SSD, and with 32 GB RAM (as much as the motherboard will take). It's almost like going back to when I first played in 1.2 when there was absolutely no lag and far fewer system overheads for the Mac to contend with, and I can have the render distance set to 24 chunks if I want (though I usually keep it at 16). The GPU is the standard Intel motherboard job, but I know I have a spare slot to shove in a new graphics card if Minecraft starts to slow down again - that should give it a boost.
The minimum requirements is 10.9. Could you give upgrading a try?