First, we solve the clues, which are (slightly non-standard) cryptic crossword clues.

Clue	Explanation	Answer
a cadet earns ones taken from trip over seas	anagram(trip + seas - a)	STRIPES
crippled jedi! run!	anagram(jedi + run)	INJURED
dancing couples meet in cultured city	anagram(pairs)	PARIS
didn't see each other in the fog, it sounds like	synonym("didn't see each other"), homophone(mist)	MISSED
hush! one red tea fits to a tee	sh + i + rt, synonym(tee)	SHIRT
i heard they consumed infinity turnovers	homophone(ate), rotated infinity symbol	EIGHT
i love to run dizzily in a circle	anagram(i + love + to + run), synonym(circle)	REVOLUTION
one's apartment's measurements	synonym(one), synonym(apartment), synonym(measurement)	UNITS
satellite is honey follower	synonym(satellite), suffix(honey)	MOON
situation calls to remove head of property	estate - e	STATE
the beginning of winning and the end of strife?	w + ife	WIFE
the tide always engulfs the best	substring(tide always)	IDEAL

Then, motivated by the theme sksquared (the bride and grooms' shared initials), we must notice that each answer can have an existing letter replaced with a S or K. The way to do this is not unique, but there is only one way to do it that fits the answers into the crossing cell diagrams on the left, such that an S and K cross. In fact, the S word is always horizontal and the K word vertical. Unfortunately, there was an erratum where PARIS and STATE were interchangeable.

Horizontal word	Vertical word
EIGHT → SIGHT	PARIS → PARKS
MOON → SOON	MISSED → KISSED
INJURED → INSURED	SHIRT → SKIRT
REVOLUTION → RESOLUTION	UNITS → KNITS
WIFE → WISE	STATE → STAKE
IDEAL → IDEAS	STRIPES → STRIKES

Finally, there are two extraction mechanisms

• First, by reading from the highlighted cells in the diagrams on the left.
• Second, by taking the displaced letters and offsetting with the numbers on the right.

This gives two phrases:

• The horizontal words give INDRED and the vertical words give PIRITS, cluing KINDRED SPIRITS
• The offsets applied to the displaced letters yields SECRET KEYS