Chapter I

What Time Is It ?
In one place, it says "But the black kitten had been finished with earlier in the afternoon" and later, still in chapter one: "while Diana was washing your face this morning" so, was it morning or afternoon ?

I think Carroll is hinting to the fact that through the looking glass time is undetermined. Look at the clock in the two pictures when Alice is passing the looking glass: On this side of the looking glass the time is 15:17, at the other side, what time is it ? Both hands are almost at the same size, so, it's likely to be 15:40 but could easily be 8:16.

See remark "The Clock Through The Looking Glass" too.

"I'm saving up all your punishments for Wednesday week"
According to Martin Gardner, the story takes place on November 4th, 1859. This day was Friday so it's 5 days in the future. Is the decision of 5 days random ? I don't think so. At the end of chapter II the red queen is measuring and marking 5 yards, Alice is moving 5 times to become a queen, and what happens than ? She shakes well the red queen i.e. the black kitty. So, after 5 moves for the 5 days Kitty is really punished by shaking.
"I should have to go without fifty dinners"
As noted above, if it's November 4th now, 50 days forward brings us to Christmas. Coincidence ? I don't think so...
"Well, it WAS a nice check"
Alice is describing the red's move 6... Kt-e7+ (and fork on the queen) that could win the game for red (Alice in this side of the looking glass). But then  "that nasty Knight, that came wiggling down" 7. Kt X e7.
The Clock Through The Looking Glass
When we look at the clock in the two pictures where Alice is going through the looking looking glass, the back of the clock looks like a clock by itself, at noted above in "What Time Is It ?" remark, but:

  1. Only one hand moved to the other side.
  2. the mirror effect is not only right- left but up- down too.

So, together with Alice's move forward through the looking glass all 3 dimensions are moving.
What Season Is It ?
At her room Alice notes about the cold outside and wonders if the fire is as worm at the other side of the looking glass. When she moves there, the first thing she checks is whether the fire is warming and find out the it does. However, when she goes out to the garden and it's way there seems to be no problem with the temperature or whether, as if it's spring... Did the seasons change too through the looking glass ? In that case, is it that the house itself is the mirror when you see Autumn through the window but Spring via the door ?

"then she floated on through the hall"
If we think of the gravitation that enables such a thing, of course it must be only a fraction of the gravitation on earth. But, where is the source of that gravitation ? It can't be the ground/floor since in that case, Alice wouldn't continue floating once she came to the hall. I think the source of the gravitation was the garden (link to wonderland ?)- When she's up the stairs it's down, but once she reaches ground level it's horizontal through the door.