している is a conjugation of する. It means that the thing you do is something you do and continue doing, instead of just an instant action. Instead of explaining the whole grammar point, I’ll redirect you to this article which explains it very nicely. http://www.guidetojapanese.org/learn/grammar/teform

The のだ thing: When you want to say something explanatory you can add a の in the end instead of です/だ.
「今は忙しいの。」 = “The thing is that (I’m) busy now.”

The の can sound a little soft (feminine) so you’ll often hear people say のだ instead. Sometimes you’ll see a んだ instead, probably because it is slightly easier to say.
Then I just wonder why it is corrected to のだ/んだ because I remember reading that you don’t use だ in questions, just の.

When asking for an explanation in English you often ask “isn’t it (blah blah blah)?” You can actually do the same in Japanese. 「blah blah blah んじゃない？」
「先生んじゃない？」 = “Isn’t he a teacher?”

You can also read about the のだ thing on this link under the heading “The 「の」 particle as explanation” http://www.guidetojapanese.org/learn/grammar/nounparticles#part5

Unfortunately, I can’t help you with the second one.

卵　を　かう　で　てんぽ　に　いく。 = “I’m buying eggs then going for a walk.” (though it’s not properly conjugated)
卵　を　かい　に おみせ に いく。 = “I’m going to the shops to buy eggs.”
卵　を　かい　に　いく　のよ = “I’m going to buy eggs.”

V-stem+に+いく = “going to (verb)”. Which is to say, “going to (a place) with the express purpose of (verbing)” – it’s not the same as the other English use for “going to (verb)” which means “I intend to (verb)”, so don’t get them confused. かう got changed to かい because that’s how the grammar works – quite a bit of grammar tacks onto the verb-stem, actually. And yeah, as Naoya said, おみせ = shop. You can just say みせ in casual speech, too.