The top clues
are each missing a synonym for "zero" at the marked point.
field which aims to k[NOW T]he details of our planet
→ EARTH SCIENCE
type of therapy that [NIX]on's opponent's initial running mate famously received
→ ELECTRIC SHOCK
positio[N ONE] in an orchestra
→ FIRST CHAIR
what joh[N ADA]ms was often portrayed as, by his opponents
→ MONARCHIST
runners in the same taxonomic class as [DUCK]s
→ OSTRICHES
it's found directly above the [ZIP]per on some tops
→ SHIRT CHOKER
a[N IL]lness caused by an allergic reaction to schistosomes
→ SWIMMER'S ITCH
These answers are alphabetized, helpfully confirming them.
The O grid:
The blanks on the right match the lengths of our synonyms for "zero". The arrows around the O hint that we need to fill the synonyms in, so they can be read word-search-style.
Assuming all cells have to be used, the P must be for ZIP; the Z must go on the left, because if it were on the right it would need to be used in whatever word uses the upper-right corner. Similarly, NIX must be diagonal starting from the upper-left.

The cell in R3C1 must then use the shared I as well. The cell in R1C2 must be part of a horizontal word, so that means NIL must be placed going ↙wards...

And now DUCK can only be fit into the grid in one way (overlapping the D in NADA).

The arrow traces out the word ENLACED.
X grid:
As hinted by "internal scramble", in the flavortext, each of the clues' answers has an anagram of CHRIST
within it (which is often represented by a "cross" -- either in the † shape, or the Greek letter chi (Χ)).

The answers can be filled in based on the direction of the arrows, so that the CHRIST anagram goes in the center. The red line then traces a path saying SCREW.
What next?
The two bits below the grids have transparent rectangles in them...
If you've solved Definition Resolution, another puzzle by noneuclideanisms, you know where this is going:

Hey, the ouer lines match some of the shapes in the title! And then when they're matched up, the boxes overlay a bunch of pigpen-cipher-like symbols... exactly as many as letters in our two answers so far.
These sets of shapes tell us how to fill the tic-tac-toe grid with letters from our answers:
DRC
ELA
WSN
If both players move...
..."X keeps circles down, initially" hints at the comic XKCD, which has a nice map of optimal tic-tac-toe moves.
If both players follow the prescribed strategy, we get the following result:
XXO
OOX
XOX
Questions:
1: The circled letters are CELS
; we'd say that each player here EXCELS at tic-tac-toe (given that they're playing optimally!).
2: So, to find the answer to the next question, we should look at the X cells.
3: Just like Santa's sleigh, this game of tic-tac-toe...

...is DRAWN.