I went and looked this up, out of interest. It's a phenomenon called the 'formal subject', which is sort of like an extra subject that occupies the space of the 'real subject' if it is postponed. In English it's 'it' or 'there' - so the sentence 'A man is in the kitchen' could be rendered as 'There is a man in the kitchen'. In Swedish, the formal subject is always 'det', it's not the same as a pronoun that takes the gender of the noun. A sentence like "Det är en meny/It is a menu" will always use the formal subject, because the alternative would be "En meny är/A menu is" which is confusing.
It took me a little while to get my head round this, so I'd recommend googling 'Swedish formal subject' for some more background!
You use "den" in place of "it" when the "it" is used to refer to a specific noun that belongs to the common gender:
"I have a dog and it is cute." "Jag har en hund och den är söt."
"My computer won't start and I don't know what's wrong with it." "Min dator startar inte och jag vet inte vad som är fel med den."
"Death takes everyone in the end. It is unavoidable." "Döden tar alla till slut. Den är oundviklig.
You use "det" in the same way to refer to nouns that are of the neuter gender. You also use "det" for the abstract dummy "it", when it doesn't refer to any particular noun:
"It is raining" "Det regnar"
"It is going to get easier" "Det kommer att bli lättare"
"It is never a good idea to drink and drive" "Det är aldrig en bra idé att dricka och köra bil"
Yes, except when it's the formal subject, which is always 'det'. It might help to remember that 'it is...' or 'there is...' in English always translates to 'det är', no matter what the gender of the subject is. So, "It is a dog" and "there is a fruit in the bowl" are "det är en hund" and "det är en frukt i skålen" even though 'hund' and 'frukt' are both 'en' words.
Through pronouns, no. Hon läser, jag läser, all the same. In English, we have more or less a simple way of conjugating. I, you, they, we read, he/she/it reads. Swedish, though, changes in the past/present/future tenses. This is when some verbs show their dark side and become what you call "irregular."