As a child I used to swallow toothpaste instead of rinsing it out. Would it have done any damaged to my body?

No. Too much flouride may cause white spots on adult teeth, called fluorosis. But no other problems.
Probably not. Most of us swallowed toothpaste when young. I've not read in any of the scientific literature that this has been linked to any problems later on in life, so far.
Not really. There is a trend in dentistry where we recommend to brush, swish and spit, but not rinse to help get the most protection with the toothpaste, the amount you swallow is not enough to cause damage.