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:

:For(I,1,137 //number of elements I bothered to enter
:If remainder(Ans,2

TI-81, #5 (67 bytes):

:Lbl 0
:If A
:If A=0
:Goto 0
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