How can I confirm my new graphics card is installed correctly?

[cscs]

So everything looks right.

When you want to use the beefier dGPU call it with DRI_PRIME=1

Special case for steam … dont launch steam itself with prime.

Instead edit each game launch options to be (assuming no other options):

DRI_PRIME=1 %command%

PS.

You have no SWAP. When you reach the limits of your physical RAM the system will lock up.
You have many options for SWAP, but I generally suggest zram these days.
Continuing the discussion from Awful performance since install:

@cscs

sudo pacman -Syu zram-generator

Then edit the configuration file. (use your preferred editor)

sudo micro /etc/systemd/zram-generator.conf

You want it to look like:

[zram0]
zram-size = ram / 1
compression-algorithm = zstd
swap-priority = 100
fs-type = swap

You may want to ensure zswap is disabled as well. One way is to add to the boot options, at /etc/default/grub the option zswap.enabled=0.
That is to add it to one of the CMD lines so that (Assuming no other options) it would look like:

GRUB_CMDLINE_LINUX="zswap.enabled=0"

Then run

sudo update-grub

And reboot.
You can check the configured swap/zram using

swapon

zramctl