1) "Trotzdem" is an adverb that is used similarly to a conjunction. Conjunctions can only be placed "between" sentences, adverbs can be placed anywhere in principle (though some placements are more natural than others, and some are extremely unnatural). This category of words is sometimes called "causal adverbs":
In a main clause, the verb must be in 2nd position, and a subclause occupies the first position "inside" a main clause, so the correct word order is:
Obwohl es regnet, gehe  ich trotzdem nach draußen.
Es regnet. Trotzdem gehe  ich nach draußen.
Es regnet. Ich gehe  trotzdem nach draußen.
"Ich gehe nach draußen trotzdem" or "Obwohl es regnet, gehe trotzdem ich nach draußen" etc. would be very unnatural, though you could run accross them in more unusual poetry, for example.
2) Yes, yes, and yes.