- No, "sig själva" is a reflexive pronoun. Själva here is an adverb that tells us they did it themselves, i.e. without external help.
- No, not really. It just doesn't make sense or sound right to me as a native speaker.
- The sentence "Han ser själv" means he is able to see without help. But "han ser sig själv" means he sees himself (e.g. in a mirror or a photograph). In English, these two uses are both "himself", but in Swedish, they're separate; "reflexive pronoun + själv" or just "själv".
Does that help?
Well, the sentence above - "de gör det själva" - tracks the English word order word for word: "They do it themselves." The other sentence you mention - "jo, det gör jag" - is still V2, it just places greater emphasis on "det" rather than "jag." One English circumlocution, if not translation, might be " yes, that is what I'm doing." You're placing the emphasis on the object, rather than the subject. So: "Ät inte hundens mat! -- Jo, det gör jag." "Don't eat the dog's food! -- Yes, that's what I'm doing."
I don't think it's quite correct to say that Jo det gör jag puts any special emphasis on det. This is the most neutral word order possible, so it doesn't emphasize anything, it's just neutral.
That's what I'm doing would be Det är det jag gör in Swedish. (put the main stress on the second 'det').