Last one first.
Do Debian based systems (Deepin 15.7) play nicely with Ubuntu based systems?
Typically, yes
. Is it the full Distro Deepin or the Manjaro with the Deepin DE? I have the latter on this Dell, and I have the former sharing with 40 other Distros on the Toshiba Satellite. I have not been into the former much, but it seems to be "happy" so far.
I never really set up LM on laptop 1, I set up MATE instead and just did a few things on LM. At this point would it make more sense to start with a clean ssd?
Up to you. If you choose to install LM fresh on the SSD (or any other Distro), that would be quickest. If so, then choose "Something Else" in Ubiquity, or Manual Partitioning in another Distro. Otherwise Ubiquity will install a Swap by default, and likely to the size of your RAM, ie 16GB. Ubiquity may also seek out and find the ESP that is on /dev/sda, currently being utilised by Ubuntu MATE, and use it to boot from.
There are Pros and Cons to this. With two physical drives, if a Distro on Drive B is booting from Drive A's ESP, and Drive A falls over or is removed or replaced, then the Distro will not start without installing an ESP on its own drive, and editing its fstab to reflect same. Vice-versa applies.
If you choose to have an ESP on each drive, then each time you install a Distro to that Drive, you want to choose Something Else, and as a part of that, select the ESP you
do not want to use and choose "Do not use this Partition". Otherwise Linux will either try to head for the nearest one available, or try to use them both and get confused.
In the case of starting the SSD from scratch, then, I would best advise you to choose "Something else" or its equivalent by Distro, Then create /dev/sdb1 as your ESP, 300 MiB or so (550 MiB if you ever want to use Solus on that drive, it insists on it), /dev/sdb2 as your Distro's / and go from there. The ESP option in your circumstances will be offered as a choice of /boot/efi and make sure it is FAT. Once created and installed, GParted will reflect it in "flags" as boot,
esp.
Cheers
Wiz