1. How is noon or midday written under the 24 hour clock?

Noon or midday is written as 1200.

Midnight is 0000.

2. How do I remember how many days there are in a month?

 

 

Remember this poem:

" 30 days hath September
April, June, November.
All the rest have 31,
except February alone which has 28." (29 in leap years)

 

3. Why are there two types of times?

The 12 hour clock system (am and pm) can be confusing especially on timetables because of the same numbers for times used twice each day.
The 24 hour clock system is often used at airports, railway stations etc.