Can you escape a grapple during a time stop (without teleporting or similar effects)?
Let's break this down:
This spell seems to make time cease to flow for everyone but you. In fact, you speed up so greatly that all other creatures seem frozen, though they are actually still moving at their normal speeds.
Describing relative time, two time frames occurring at the same time. Just that one time frame is happening much faster than the other.
You are free to act for 1d4+1 rounds of apparent time.
So essentially what this means is you're being allotted X amount of actions for 'free' in between your casting of Time Stop and the opponent's next turn.
While the time stop is in effect, other creatures are invulnerable to
your attacks and spells; you cannot target such creatures with any
attack or spell.
So really it doesn't say I can't attack or target the opponent with spells just that the results of the attack or spell would be irrelevant.
Taken from Grapple 'If you're grappling':
Some of these actions take the place of an attack (rather than being a standard action or a move action).
And under the 'Escape a Grapple' option:
You can escape a grapple by winning an opposed grapple check in place
of making an attack. You can make an Escape Artist check in place of your
grapple check if you so desire, but this requires a standard action.
So it's not considered an attack? Though if you take the position that it is an attack for the purpose of Time Stop then there's no way to escape a grapple by making an opposed grapple check because your action is irrelevant to the opponent under the invulnerability clause noted above. So how about Escape Artist, that's definitely not an attack it's a skill check that takes a standard action.
Further support for the postulation making a grapple is not an attack taken from 'Grapple checks':
Repeatedly in a grapple, you need to make opposed grapple checks
against an opponent. A grapple check is like a melee attack roll.
So 'like' a melee attack roll not 'is'.
Now I'm picking up some impression from the commentary that there's some issue about the opponent's ability to make the opposed grapple roll. However the opponent is not taking actions, not attacks, not doing things that take the place of attacks. The opponent is responding to the player's actions necessary to resolving the grapple by making an opposed roll. Under normal circumstances he would be required to do this for however many grapple check that require resolution, the only upper limit is dictated by how many characters can be involved in a legal grapple and exercise the option to grapple the opponent.
In addition taken from the spell description and reproduced above 'In fact, you speed up so greatly that all other creatures seem frozen, though they are actually still moving'. So it's not as if the opponent is 'frozen in time' simply that that the opponent is opposing the grapple at a different time frame reference to that of the character.
Finally, as a standard of comparison, the haste spell somewhat mimics time stop in the effect it exhibits but to a much lesser degree in that a character is acting slightly faster than opponents in the same time frame of reference. Under normal circumstances with a character under the effect of a haste spell would an opponent be required to make an opposed grapple roll in response to the character's grapple attempt? The answer is yes.