First, let's analyze all three constructions:
- Two consecutive verbs, both in imperative mood.
- A verb in imperative mood plus infinitive.
- Two verbs in imperative mood separated by
Second form is actually a special case of composite predicate (составное сказуемое), which can consist in Russian of verb plus another verb in infinitive.
Compare "идти работать"/"успеть долелать" with "иди работать!" and "успей доделать!". Or "люби вкалывать". Or "заканчивай ныть".
With such composite predicates the logical center is the first verb.
See, you can say
вкалывай, but you'll loose the meaning of composite predicate.
Ной is even more striking example, since it turns out to be an exact opposite of "заканчивай ныть".
But the verb
идти is different. It can be used very similar to that how the verbs
come are used in english imperatives. This is verb is kind of amplifier of whole imperative construction. Just like in English we can say "Go and buy us some food", we actually mean that on should just buy food. Or "Come find me" which can be translated either like "иди найди меня" or just like "найди меня".
To conclude: Second form mean the same thing as first form and third one. Third form has a subtle pathetic connotation in Russian (Compare "иди и смотри" and "иди смотри"). Coexistence of "imperative + infinitive" and "impertative + imperative" forms is due to ambivalence of "идти" which can be used both as part of composite predicate and as imperative amplifier (the last term is not strictly linguistic).
Also, in some languages there exists so called deferred action imperative (императив отложенного действия), which is expressed, among other forms, as two consecutive imperatives: one for first action and one for action which follows.
As a rule of thumb: if you can insert
then, you can use first and third forms.
"иди смотри"/"Сначала иди, и потом смотри", "возьми и сделай"/"сначала возьми [соберись, приведи себя в порядок], а потом сделай".
I've never heard term "императив отложенного действия" in context of Russian language, but, as for me, examples you've provided can be considered as some special form imperative "specie".