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": http://www.canoo.net/services/OnlineGrammar/Wort/Adverb/Klasse/Kausal.html?lang=en 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 [2] ich trotzdem nach draußen. Other possibilities: Es regnet. Trotzdem gehe [2] ich nach draußen. Es regnet. Ich gehe [2] 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.

To elaborate some more on Q1: Some people use "trotzdem" as a conjunction equivalent to "obwohl": "Trotzdem es regnet, gehe ich nach draußen." That's not (yet?) considered correct in standard German, though, hence the remark in Collins.