"и" is, sometimes, used in Russian as the emphasis and sometimes could be skipped without losing the sense of sentence. In that case it is translated as "that is what, that is were, that is who"

examples (from Lingvo dictionary - strongly advise you to get some at GooglePlay or AppStore)

"здесь он и жил" - "it was here that he lived"

"вот об этом он и думает" - "that is what he is thinking of"

"вот и всё" - "that is it" 

so in your example

"мне довольно и того, что.." almost equals  "мне довольно того, что", with the sense

"for me only that is enough, that" vs "for me is enough, that.."

the first example is trickier - "того" ("того" is Genitive m. and n. forms of "тот" - "that")

"он лёг отдохнуть, да и того" - is tricky, because you can ask - "и чего, того?", "а того, что помер" ("he laid down to rest, and that was it", "that was what", "he died").

"он это - того" - "he is - well, crazy or dead" - depending on the context

"ты что совсем того?" - "are you crazy? (what the f** are doing?)"

1. Того или Этого = It, that

2. Того is a fun  word and we use it in all kinds of situations when we don't remember an appropriate word or don't want to say the word or are too lazy to say it. So you have to pretty much guess what  того means from the context or the situation. For example A person is behaving in a strange or inappropriate way and we say: Кажется, он того.. meaning that he's kinda crazy.

In your first  sentence того means that he fell asleep.

In the other one: It's enough for me that you're next to me. Того means it