Step by step deduction
First look at the right middle box, since that's got four cells filled already including two odd in a column and two even in a column.
Also, from normal Sudoku rules,
Now look at the left middle box, which also has four cells filled already.
The $1$ in the right-middle box can now be filled by normal Sudoku ...
First, some preliminary steps:
Now, we begin the actual deductions.
Next we look at a certain square:
And now it's time to finish off the assignment:
The top right can be resolved: first consider where the first square in the second row goes, then how the square above-right of it is accessed.
Next, the top left: the key step is figuring out ...
We start off by placing most of the $1$ and $2$ cells across the whole board, using all the given conditions to narrow down the possibilities. They can both be placed in the left-middle, centre, and right-middle boxes right away, and then we can make some more deductions in the upper and lower boxes:
Now a slightly ...
There are five possible tetrominoes: L, T, O, I, S. There are four different symbols, giving six possible pairs of distinct symbols (blue-green, blue-yellow, blue-red, green-yellow, green-red, yellow-red). So one of the pairs doesn't occur, and the other five have a one-to-one correspondence with the tetromino shapes.
Glorfindel's method of assuming uniqueness of solutions is OK and valid (for Sudokus), but a puzzle with a unique solution can always logically be solved without resorting to such logic. Here's how I did it.
First, you should look for a cluster of cells which have few possibilities (ideally just two) and are closely related, so that if any one of them is ...
For the maximum sum,
I saw SteveV's answer and noticed that he wasn't
With that in mind, I came up with a new solution.
For the minimum sum,
Here is my solution for the minimum sum
I haven't been on here lately and have kinda forgotten how to format so will edit while I figure things out
Start from the bottom:
Now consider the 3-letter words beginning with P.
Now in the top right we have limited possibilities:
Now there's lots of other stuff to be filled in around those long words on the left, and some other deductions which lead us all the way to the centre.
Now only four 5-letter words remain to be used:
Step by step:
(This is the first time I've tried to solve one of these, so sorry if my explanations and/or logic are confusing or in a weird order and if my terminology is weird :) )
Final State (Solution)
My progress (images that I saved "in the middle"):
First few steps
First (and only) time I made a guess (See the bottom for the way of avoiding this guess, making this answer completely logical)
The exact guess I made was
After making that guess, continuation was pretty straightforward: You stare at the board, following regular ...
The grid looks like this:
Making the missing word
And listing the words:
Explanation of how I got the grid.
I know this logic is flawed as the missing word could fill in, but that was my thought process that got me the answer so I guess it works here :)
Full credit to @AxiomaticSystem for this answer (2nd problem). I'm just writing my personal logical deductions. Apologies for how long this is; I can make deductions but struggle to explain them succintly.
That was long. But there you have it: logical deductions for problem #2.