Why is September the ninth month when it means 7, and likewise for October, November, December...?

Because the calendar has changed over time. There weren't always 12 months in the year. The months got named mostly during Roman times--hence the Latin roots for their names--but not ALL did--and the Roman calendar was a lot stranger and weirder than people realize. 

When the months were named, they were named for their positions in whatever calendar they were using then:  September, October, November and December were the 7th, 8th, 9th and 10th months then. July was named after Julius (Caesar) and January after the 2-faced Roman God, Janus. Those are the ones we still use, despite what the names mean. 

The calendar of 12 months we now use is the Gregorian system--a year of 365 days divided into 12 months. As you know, it's not perfect, because there are actually 365 and a FOURTH day every solar year--so every four years we end up with an extra day--which we tack on to February and call "Leap Year."  But it's only a little more accurate than the one the Romans used (The Julian Calendar) because that one did not align with the events such as solstices and equinoxes. When the Gregorian calendar replaced the Julian calendar 11 days were "lost" (meaning not counted, not actually lost.) 

Though the Gregorian calendar has been in use for over 430 years and is considered the most widely used and "standard" way of marking the year, it is not the only calendar in use. Jews use a different calendar, as do Asians, Muslims, Indians and other countries and the Catholic church. Some of them are completely ridiculous and so complex they boggle the mind. The Mayans used TWO different ones! 

Ever hear of Julius Augustus Caesar? Well ... he wanted to be remembered forever ... and not just for his salad.

Same reason the shortest month is Februari: The Romans had their years start March first

It all derives from the original Roman calendar, when there were 10 months.

10th month was December - Decimal mean anything to you ? number system based on 10. Decimus

Likewise the other names derive from the roman numbers 

Same reasons why the order of the seasons are Spring, summer, fall winter.  The year use to start in the spring.  It all makes more sense.  Now they year starts and ends in the middle of winter.

Even why February is the shortest month and leap year adds a day onto that month.   Because is was the last month of the year.

The reason is that the starting point of the year used to fall in March.

Original Roman calendar was 10 months, but it was always a bit messed up as it began in March.They added January and February  and July (after Julius Caesar) and August (August Caesar) never followed the numbering logic - so the wheel came off the numbering scheme pretty much from the outset.

September was originally the seventh of ten months in the oldest known Roman calendar, which eventually developed into the calendar we use today.