Omnispell Adept and Alternative/Additional costs of Sorceries/Instants
Alternative, no; kicker: yes. The rulings below the card in Gatherer state so:

If you cast a spell “without paying its mana cost,” you can’t choose to cast it for any alternative costs. You can, however, pay additional costs, such as kicker costs. If the card has any mandatory additional costs, such as that of Severed Strands, those must be paid to cast the card.

Basically, Omnispell Adept's ability is already an alternative casting cost and you can't replace that. The relevant rule is 117.9:

117.9. Some spells have alternative costs. An alternative cost is a cost listed in a spell’s text, or applied to it from another effect, that its controller may pay rather than paying the spell’s mana cost. Alternative costs are usually phrased, “You may [action] rather than pay [this object’s] mana cost,” or “You may cast [this object] without paying its mana cost.” Note that some alternative costs are listed in keywords; see rule 702.
117.9a Only one alternative cost can be applied to any one spell as it’s being cast. The controller of the spell announces their intentions to pay that cost as described in rule 601.2b.