The "t" in "det" is silent, and the vowel short, kind of like "deh" or like the English "day" but with a much shorter vowel sound
"De" is pronounced like it should be written "Di" in Danish, with an "ee" sound, like the name of the letter D in English, but with more pressure on the vowel. Sorry if this isn't clear, but that's the best way I can describe it. The voice can make the words flow into one another sometimes, which isn't necessarily wrong, but it probably does make it harder to hear the individual words properly while learning.
Yep. "I" can sometimes be used to mean "this" when talking about time.
Here are some examples:
I dag (today)
I morges (the morning of today, or last morning if said during the morning)
I morgen (tomorrow)
I overmorgen (day after tomorrow)
I går (yesterday)
I foregårs (day before yesterday)
I år (this year)
I aften (this evening/tonight)
I nat (this night/tonight)
However, this construction does not work with other words like: Måned (month) or Uge (week).