Now that the contest is over and I can figure out how to implement these algorithms, I'm going to see what I can come up with.

I made two solutions for #5, for TI-73 and TI-81. Neither have a proper string type, so the TI-73 takes input from L₁ and the TI-81 takes partial input from {x} (it doesn't have enough memory to hold the entire #5 prompt).

TI-73, 58 bytes (add 6 for VAT stuff and one character name) and probably simple to beat:

Code:
:5->X
:5->Y
:For(I,1,137 //number of elements I bothered to enter
:Pxl-On(Y,X
:L₁(I
:If remainder(Ans,2
:Then
:X+Ans-2->X
:Else
:Y+Ans-3->Y
:End
:End


TI-81, #5 (67 bytes):

Code:
:9->X
:Ymax->Y
:1->I
:Lbl 0
:PT-On(X,Y
:{x}(I->B
:(-1)^B+1->A
:If A
:Y+3-B->Y
:If A=0
:X-2+B->X
:IS>(I,Dim{x}
:Goto 0
  
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