The ground screw business is...tricky.
We start with the applicable Code reference, namely NEC 250.146 incl. (A) and (B):
250.146 Connecting Receptacle Grounding Terminal to
Box. An equipment bonding jumper shall be used to connect the grounding terminal of a grounding-type receptacle
to a grounded box unless grounded as in 250.146(A)
through (D). The equipment bonding jumper shall be sized
in accordance with Table 250.122 based on the rating of the
overcurrent device protecting the circuit conductors.
(A) Surface-Mounted Box. Where the box is mounted on
the surface, direct metal-to-metal contact between the device yoke and the box or a contact yoke or device that
complies with 250.146(B) shall be permitted to ground the
receptacle to the box. At least one of the insulating washers
shall be removed from receptacles that do not have a contact yoke or device that complies with 250.146(B) to ensure direct metal-to-metal contact. This provision shall not apply
to cover-mounted receptacles unless the box and cover
combination are listed as providing satisfactory ground
continuity between the box and the receptacle. A listed exposed work cover shall be permitted to be the grounding
and bonding means when (1) the device is attached to the
cover with at least two fasteners that are permanent (such
as a rivet) or have a thread locking or screw or nut locking
means and (2) when the cover mounting holes are located
on a flat non-raised portion of the cover.
(B) Contact Devices or Yokes. Contact devices or yokes
designed and listed as self-grounding shall be permitted in
conjunction with the supporting screws to establish equipment bonding between the device yoke and flush-type
If you're using a surface-mounted gang or "handy" box, complying with 250.146(A) is easy as the device yoke is mounted directly to the box, with the faceplate in turn mounted to the device, just as you would with the boxes you typically use in flush work. However, square boxes (100mm and 120mm) are annoying in that they don't have mounting screws for receptacles. You either need to use an adapter known as a mud ring to provide a single gang opening that you can then mount the device yoke to conventionally, or you need a flat cover listed to be a grounding and bonding means in exposed work, with the receptacle mounted to it using at least two permanent or locked-thread fasteners that go through mounting holes in the cover and into/through the device yoke.
Given all this, it's probably simpler to put a ground pigtail in for the device -- it's cheap, and avoids issues with the GFCI not having a ground to offer when it's not all buttoned up, or the cover isn't seated properly on the box (which can happen!). Either that, or use a single gang mud ring instead of the 4" cover -- you'll gain a bit of volume by using the mud ring, too.
Other than that, you have it right on the money
Once you take care of grounding the GFCI, the rest is as you describe: pigtail the two load-side hots together to the LOAD hot terminal, and the two load-side neutrals together to the LOAD neutral terminal.