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Yes, the Demilich can use Astral Projection, even on himself. Infact, from reading the spell you HAVE to go with your intended travelers if you want to use the spell at all.

As my other answer clarified, "Magic Immunity" is not the same as "Spell Immunity," and while I would argue that they are nearly identical that doesn't matter for the Astral Projection SLA as it's own definition is enough to allow the spell.

As I stated in my other answer, the definition of "Magic Immunity" is (Emphasis Mine):

(Creature) is immune to any spell or spell-like ability that allows spell resistance. In addition, certain spells and effects function differently against the creature, as noted in its entry.

This clearly states that magic immunity is not 100% set in stone. This is what allows the Demilich to use it's 'At Will' Harm SLA on itself. Since both Harm (which explicitly states it is mainly used to 'heal' itself) & "Astral Projection" (again, normal casting requires the caster to be the primary target) are listed in it's SLAs then RAW they are able to be cast on itself. Someone might ask why it didn't explicitly state how this spell would work on the demilich, the answer is simple, "it doesn't have to." The only time it HAS to list how something interacts with something else is if that interaction is different from normal. So, had the writers intended to allow the Demilich to cast the spell without it being the primary target (& avoid traveling) THIS exception/variation would have had to have been stated. However, listing the spell without modifications means simply that it was to be cast as usual.

Some may ask, "was it simply an oversight?" And while none of us can disprove a negative, I can state they WotC deliberately & explicitly removed it's "Alter Self" SLA ¹ (which ironically doesn't require Spell Resistance) but left this spell! The fact that exists in the first place is enough to infer that it was intended to be usable, but the fact that the definition explicitly allows it cements the usability in the affirmative. Hence meeting the RAW requirement.

Bottom Line: The rules explicitly (maybe not too clearly) allow the Demilich to use it's Astral Projection SLA on itself.

¹ The SRD is not updated to reflect that "Altar Self" has been removed from the Demilich's SLA, but according to the errata dated 2/6/2006 it has been removed.

Summon Monster doesn't allow you to summon any creature of the elemental subtype, it allows you to summon the specific creature called “Elemental”. The table is a list of specific creatures, not creature subtypes:

The spell conjures one of the creatures from the 1st Level list on Table: Summon Monster.

What might be throwing you off is the links in the Summon Monster tables in the PFSRD. Unfortunately, the links are misleading. Most of PFSRD's links are automatically detected, but the generator can't tell the difference between “elemental” (the monster name) and “elemental” (the outsider subtype). Since the subtype is much more commonly what that string of letters means in the SRD, the automatic linker defaults to the page for the elemental subtype — which is the wrong meaning in this case.

In other words, when it says that Summon Monster II can summon an “Elemental (small)”, it means that you can summon a Small Fire Elemental, a Small Lightning Elemental, etc., not a small version of any creature with an elemental subtype.