1. What is the plural of index?

The plural of index is indices.

2. What are the other names for an exponent ?

An exponent is sometimes called a index or power.

3. Why is a dot sometimes used to mean "multiply" in algebra?

One of the reasons could be that if a normal multiply sign was used if could be confusing when using x as a variable.

e.g. x × means x 'multiplied by' x. It is clearer if written x.x

Note it is NOT a decimal point.