If I used the -8 ability of Jace, Architect of Thought to search for and cast a MDFC from my library, like Valki, God of Lies, do I get to choose the side that I want to cast? Would the same be applied if I cast it for free using Omniscience?
In general, as you cast a modal double-faced card, you can cast either face (not: side) of that card, and "[o]nly the face that will be face up on the stack is evaluated to determine if it can be cast" (C.R. 712.7, 712.7a). Thus, for example—
you can cast the back face of that card with Jace without paying its mana cost, if you exiled it with Jace's last ability, and
you can cast the back face of that card without paying its mana cost with Omniscience if the card is in your hand.
And in general, you can cast Tibalt, Cosmic Impostor (the back face of Valki) with Jace's last ability or Omniscience without paying its mana cost.
However, an effect can allow you to cast only spells of certain kinds in a special way. One example is the recent rule change to cascade, where you can cast a card using cascade only if "the resulting spell's mana value is less than" that of the spell with cascade (C.R. 702.85a); see this thread. Another example is Kari Zev's Expertise; see this thread.
Thus, with Kari Zev's Expertise or Bloodbraid Elf's cascade ability, you can't cast the Tibalt face, since that face's converted mana cost is too high in either case.
EDIT (Mar. 29): Clarification.
EDIT (Nov. 12): Edited, including because some rules were renumbered in the meantime.
And in general, you can cast Tibalt, Cosmic Impostor (the back face of Valki) with Jace's last ability or Omniscience without paying its mana cost.
However, an effect can allow you to cast only spells of certain kinds in a special way. One example is the recent rule change to cascade, where you can cast a card using cascade only if "the resulting spell's mana value is less than" that of the spell with cascade (C.R. 702.85a); see this thread. Another example is Kari Zev's Expertise; see this thread.
Thus, with Kari Zev's Expertise or Bloodbraid Elf's cascade ability, you can't cast the Tibalt face, since that face's converted mana cost is too high in either case.
EDIT (Mar. 29): Clarification.
EDIT (Nov. 12): Edited, including because some rules were renumbered in the meantime.