Integers are whole numbers and are used constantly, especially when, before I retired from work, I would check that my wage package (wage slip) was correct! So while 1, 2, 3, or -1, -2, -3 are integers, the decimals 11.2, 234.55 or -11.2, -234.55 are not.
Every time you need to use 1, 2, 3, and so on in maths, or even counting your money to ensure you have enough to get the bus home rather than having to walk, you are using integers!
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Every aspect of mathematics depends upon integers, and even if you don't do any mathematics, you count things with integers. If you even want to know how many guests you are having for dinner, you need to use integers.
You can sometime use integers but often speed or velocity can also be irrational.
Yep.
Yes.
You can use them for counting, which is in fact addition. You can also use them for subtracion. Integers are natural numbers. This means you don't use decimals or fractions.