1) skating across the pond
2) studying for weeks
3) correct.
4) writing letters at camp
5) finishing the dishes
6) greeting the guest
7)correct
8) this was more complicated but I think it's correct
9) watching television
10) lost in a strange city
Okay, so these are actually really easy once you get it. Basically, if the noun after the comma isn't what was doing the action then it is wrong (unless it was stated before the comma) for example
”Forgetting to do homework, the paper was turned in late.”
That's wrong because it makes it seem like the paper was the one who forgot to do their homework. You could fix it by saying
“forgetting to do my homework, I turned the paper in late”
Hope I helped!