Can a DNA test determine if a mother had multiples?

Testing the mother or one of the children will not reveal whether other children were born from the same pregnancy.

DNA testing on multiple children can establish that they are identical twins or siblings. If the testing reveals that three of them are identical twins, then they were triplets.

For non-identical twins, it's more complicated. If the testing reveals that they are siblings, there is no way with a DNA test to distinguish between non-identical twins from the same pregnancy and siblings born from separate pregnancies. If the children are very, very young, you can determine their age well enough to know whether they must have come from the same pregnancy (or must not have come from the same pregnancy). As the children get older, it is harder to tell their real age for sure through medical examination, so a 15 year old girl could find out that she was really a non-identical twin of her 14 year old sister (and that she's not really 15). If the women are 58 and 59, there is no way to examine the 59 year old and find out that the 59 year old is really 58 years old (and thus must be a non-identical twin if the DNA tests confirm that they are siblings).

No, it wouldn't.

yes, but only if the multiples all had the same dna test.