A Bbbbusy Day A Bbbbusy Day

I assume the extra “bees” in “busy” aren’t a typo. :-)

A Bbbbuuuusy Day A Buuuusy Day

If you put multiple consonants it sounds like a stutter ("b-b-b-b") Some consonants are okay to do it with, like "n" and "m" (also the "sh" sound) but generally we only draw out vowels to emphasize a word

A Bbbbusy Day A Busy Day

The meeting place was quite close to my home, butuse; it was on the other side of thea river. The meeting was quite close to my house; it was on the other side of a river.

I took a taxi, but the traffic was so terriblebad that I barely arrived on time for the meeting. I took a taxi, but the traffic was so bad that I barely arrived on time.

Now, it's half past 11 at night. Now, it's half past 11 at night.

Or you can say “now, it’s Half past 11pm”

Now it's half past 11 at night. Now it's half past 11 at night.

Your wording is fine. You could also say it’s almost midnight.

I'm tired, but I can't go to sleep immediately because I drank three cups of coffee this morning, and the gear for tomorrow's hike hasn't been packed yet. I'm tired, but I can't go to sleep immediately because I drank three cups of coffee this morning, and the gear for tomorrows hike hasn't been packed yet.

I think you can say both tomorrows and tomorrow’s, but I think it depends on context

I'm tired but can't go to sleep immediately because I drank three cups of coffee this morning, and the gear for tomorrow's hike hasn't been packed yet. I'm tired but can't go to sleep immediately because I drank three cups of coffee this morning, and the gear for tomorrow's hike hasn't been packed yet.