So I have a creature that's been mutated twice, let's say Cold-Eyed Selkie with Gemrazer and Chittering Harvester (on top). I cast Suspend on the creature. Suspend's wording is "When the last (time counter) is removed, they play it without paying its mana cost. If it's a creature, it has haste."
Can I then play it without paying its mutate cost and bring the Cold-Eyed Selkie back in with Gemrazer mutated and Chittering Harvester mutated onto it for free?
Mutate expresses an alternative cost that's paid rather than the mana cost (C.R. 702.140a). Since the suspend keyword has you play a card without paying its mana cost (C.R. 702.62a), you can't apply any other alternative costs, such as mutate, to playing that card this way (C.R. 118.9, 118.9a). (Note that to play a card means "to play that card as a land or to cast that card as a spell, whichever is appropriate" [C.R. 701.14b].)
In any case, if you exile a mutated creature with Suspend, then all its components will be exiled and each of them will be given two time counters and gain suspend (C.R. 723.3c, 702.140c). (Then any of those components that are tokens will cease to exist as a state-based action [C.R. 704.5d].)
Can I then play it without paying its mutate cost and bring the Cold-Eyed Selkie back in with Gemrazer mutated and Chittering Harvester mutated onto it for free?
In any case, if you exile a mutated creature with Suspend, then all its components will be exiled and each of them will be given two time counters and gain suspend (C.R. 723.3c, 702.140c). (Then any of those components that are tokens will cease to exist as a state-based action [C.R. 704.5d].)