It seems that tin, dinmek, dinlemek, dinlenmek are all ultimately related, at least according to Nişanyan's etymological dictionary.
He says that dinlemek (listen) is from old Turkish tıŋla- "kulak vermek, dinlemek" ("give ear", hear, listen) which he derives from tınıġ "nefes, soluk" (breath, soul) which in turn is from tın- "solumak" (to breathe). (Not sure where the leap from "breathe" to "hear" comes from....)
And dinlenmek (rest) he derives from tın/tıŋ "nefes, soluk" (soul, breath), with a note that "Tüm Türk dillerinde tıŋla-/diŋle- fiili "kulak vermek" anlamı kazanırken, sadece TTü diŋlen- başka bir anlama evrilmiştir." (In all Turkic languages, the verb *tıŋla/diğle-" gained the meaning "hear"; only Turkish-from-Turkey evolved a different a different meaning(?))
dinmek (abate, quieten, stop) is also from old Turkish tın- "1. soluk almak; 2. dinmek, dinlenmek" (1. breathe in; 2. quieten; rest). This makes a reasonable amount of sense to me - going from "breathe" to "stop to catch one's breath" to "rest" and "quieten".
And finally tin (soul, spirit) is straight from old Turkish tın "can, nefes" (soul). The initial t- is a bit suspicious as it should have been d- (din) if it had evolved normally; Nişanyan gives the note "TTü kullanılmayan bir kelime iken Dil Devrimi döneminde Kaşgarî'de bulunarak dolaşıma sokulan sözcüklerdendir. TTü biçimin normal ses evrimi çerçevesinde din olması gerekirdi." (While this is not a word that was used in Turkish-from-Turkey, it is one of the words that snuck in during the period of the Language Reform as (supposedly) having currency in Kaşgarî. The Turkish-from-Turkey form should have been din in the framework of the normal sound evolution.)
Challenges are a good thing. :-) I've started to do the lessons very slowly and thoroughly read the comments for every sentence, regardless if I got it right or wrong. The important thing for me as a beginner, I think, is to learn the structure of the language, and not necessarily useful everyday phrases.