If I boot from a live USB image for an install can I count on the devices listed in the install menu to be the same as they are on the existing system? For instance, if /dev/sda1 is mounted at / & /dev/sdc is mounted on /home/me can I count on the same devices names (sd#) to be pointing at the same physical drives in the install menu as they are on my desktop when booted into it's native OS?

You cannot be sure they will even stay the same from one udev version to the next. I recommend using the UUID or giving them a label and use that. (I do both: the label is often easier to detect and remember when using a live media boot.)

Nope.

The order depends on which one responds first, and which controller responds first. I've had one disk jump around whenever I add or remove a different one.

the only way to easily identify the disk is by using the /dev/disk/by-id. Then entries there (other than the WWN references to support fibre channel) are identified by controller-manufacturer,model, and serial number. These entries are defined by the manufacturer (so some parts may be missing), but have so far been unique to the drive (so far, in that the only more generic one is for a DVD, identified only as ata-PIONEER_DVD-RW_DVR-116D)

The most consistent entry is /dev/sda - because that is the one the BIOS uses to boot the system, which happens to cause it to be spun up and ready before the kernel even starts, thus it responds first.

All true. However ...
Given that this is related to your other current thread, it is unlikely the devices will change name on a simple upgrade.
I would think Mint would use UUID by default.

I have all the devices UUIDs but the problem is that the mint installation screen doesn't show you the devices UUIDs, it only shows their sd#.

Should be able to tell the difference using partitions and/or size. Else keep a copy of "lsblk -f" handy.