Buy a plane ticket way in advance or right before leaving, which is cheaper?

I can take a vacation any time, so I'm just looking for the lowest price I can find. In general, is it better to buy a ticket 3-6 months before or a week before you go to get the lowest prices.

My thinking is that if you wait until the last minute, an airline with empty seats on a plane will dramatically reduce it's ticket prices to fill those last few seats.

Does this ever happen?

Is it normal for airline departure time to keep changing?

Me and my family purchase tickets from American Airlines for July 4. When we brought The tickets the departure time was suppose to be At 6:00am. It done change 3 times already now it's 7:10am. Is it normal for the time to keep changing?

Infants on Norwegian airline?

My father in law is taking us to Florida on Norwegian air premium economy and since my daughter is under 2 she doesn't have her own seat. What about meals? We get meals but do infants?
Can we order a kids meal on th flight or do we need to bring food for her?

Taking a tiara through airport security?

I'm going on a flight tomorrow and i have a metal tiara i need to bring with me because all i have is carry ons. Will this be allowed? Will i need to take it out of my bag when i go through security or will I be able to leave it in my bag?

How fast the distance between the two aircraft?

An aircraft is 144km east of an airport and is travelling west at 200 km/h. At the same time a second aircraft at the same altitude is 60km north of the airport and travelling north at 150km/h.