"Barnet stod og rørte ved kaninen."

No, although you'd probably assume that's what the person saying this actually meant to say. If it was petting you'd say either "stod og kælede med kaninen" or "stod og aede kaninen" ("at ae" means to stroke or caress).

the 'stod og' actually means 'stood and'/'was standing and'. I would say that this can be left out if you're in doubt. We say it, but usually it's not of importance in the sentence as much as if you were to say 'the kid stood and touched...' in english. (The latter more stresses the fact that the kid could do two things) In Danish, you'd just say it so quickly that it's out of importance.

We also use "sad og ..." ("was sitting and...") and "gik og ..." ("Was walking and...") (all of them also works in present form). And while they are all referring to the actual situation/activity of the subject (with no exchanges like you sometimes see in e.g. Dutch who also has this - So the kid here WAS actually standing while touching), they are mostly used just to get the picture straight: The kid was there, he was positioned like this while he did that.

But if you're in doubt whether to use it or not, just leave it out :) (However, I don't know how DL feels on that).

I don't know any. But without further investigation, I would think that 'er ved (inf)' goes every time, also where you could also use 'sidder' or 'står'... For instance if you are cooking you would say 'jeg står og laver mad' or you could just as well say 'jeg er ved at lave med'. Homework: 'Jeg sidder og laver lektier' or 'jeg er ved at lave lektier'. Just notice that the verb changes from present to basic form (or what it's called).

Yes. Hund is of Germanic origin, think "hound". Kanin comes from the Latin word for rabbit, "cuniculus". It may be related to the English "bunny", but that's disputed, and honestly not very helpful. :´)

We use the -ing form all the time in English, even for just a regular present tense. ( I am walking, etc.) In contrast, with Danish, you ordinarily use just the regular present tense Jeg går. But if you really, really want to emphasize that you are doing something RIGHT NOW, you can use one of these progressive constructions.

When you get the chance, check out the web version of the exercises. Before choosing the lesson, you can read the blurb about the progressive aspect of Danish. They don't used anything analogous to the -ing ending, a feature which may be unique to English (out of Germanic based languages at least).