Yes, in casual speech पैर can mean both feet and legs.

That is an acceptable translation as well.

Yes. Generally speaking, हाथ (hand) can also mean arm (बाँह), forearm (कलाई), and palm (हथेली).

That’s तैरना which means “to swim”

I think because the literal phrase is "Two hands and two feet are close to me." Since the subject is feet and hands and it is plural, we use हैं

See this link: http://www.learning-hindi.com/post/3296383844/lesson-92-%E0%A4%95-%E0%A4%AA-%E0%A4%B8-ke-paas-to-have

@zeebo7 answered this on another question Are you maybe typing ो as ा+े? He was right. That's exactly what I was doing with the 'o' in do. That's why I got it wrong.