"I am lying alone and reading."
Translation:Я лежу один и читаю.
It really depends on the adverb. Some adverb work better after the verb, others before it. I suggest you just learn them one-by-one.
There is no real rule, just guidelines. I've translated the guidelines about the adverbial modifiers of manner here: https://www.duolingo.com/comment/11620891
Сам is an emphatic (or intensive) pronoun, like "myself" or "himself" in English. (In English, these are similar to the reflexive pronouns, btw, although in reality they are separate: "I hurt myself today" = reflexive, whereas "I will do it myself" = intensive/emphatic.)
Hence, 'сам' in Russian doesn't mean alone, which is 'один' (m) or 'одна' (f).