I know that someone can't activate shallow grave unless both players have a monster in the graveyard, but does the same rule apply for a Gilasaurus's special summoning effect? I know Rex did it without monsters in Joey's grave, but we all know how the show misdirects us sometimes/a lot of the time!

Gilasaurus does NOT require a monster to be Special Summoned...that effect is completely optional to the opponent, unlike cards like The Shallow Grave which DO require both players to Special Summon.

So, you CAN Special Summon Gilasaurus if the opponent has no monsters (or legal monsters) in their Graveyard to Special Summon.

yes you can special summon gilasaurus without your opponent having a monster to special summon

And, yes, Skill Drain does prevent the opponent from Special Summoning from Gilasaurus' effect, even if they want to (since Gilasaurus' Special Summoning for the opponent takes place when it's face-up on the field, which Skill Drain negates. Its own Special Summoning from your hand is not negated since Skill Drain does not affect the hand)

You can special summon gilasaurus even if there is no monsters in your opponent's graveyard.

In case no one believes this...

UDE/Netrep

The effect of this card only activates when you elect to change its Normal Summon to a Special Summon. Not if it is Special Summoned by "Monster Reborn", "Marauding Captain", etc.

You can use the effect of this card to change its Normal Summon to a Special Summon even if your opponent has no monsters in their Graveyard.

[Re: Necrovalley] If "Necrovalley" is on the field, you can Special Summon "Gilasaurus" using its effect, but the effect that lets your opponent Special Summon is negated.

[Re: Skill Drain] You can Special Summon "Gilasaurus" while "Skill Drain" is active, but its effect is negated so your opponent does not get to Special Summon.

[Re: The End of Anubis] While "The End of Anubis" is on the field, if you Special Summon "Gilasaurus", your opponent can't Special Summon.