Yes, you read the dice separately. So with 1 unmarked progress, your target number is 2+, meaning each die must read 2 or higher to count as a success. So yes, a 1 and a 4 is a partial with a 3 and a 5 being full. Double 1s is just bad luck.
Yes, any attribute+skill combo you can reasonably argue as relevant (players choice) can be used to make a check. So if you have Swim, and need to argue your ability, you could use your knowledge of how to swim (i.e. the Swim skill) to argue it. Skills are not tied to any specific attribute.