Asalaamu alaykum.
Semantically speaking, theres no big difference between the meaning of cult and the meaning of religion except that religion usually refers to a system based around a diety(ies) whereas a cult does not require such. (i.e. Star Trek has a cult following etc...)
Regarding dogs and not keeping them in homes - dogs are generally dirtier than cats. That's both scientifically and anecdotally proven

. Allah has decreed thats we shouldn't keep dogs, (although in the past this was allowed - see Surah Khaf). Touching one breaks wudhu - that is the main reason. Wudhu is broken by coming into contact with something najaas. Najaas is sometimes obvious and sometimes not so obvious. Since these are matters of the deen, we have to leave it to the instruction of Prophet Muhammad (pbuh) who is our teacher.
The babies hairs weight in gold given to charity is symbolic of many things; giving thanks to Allah for a blessing being perhaps the most important.
I think the most important thing to remember is that our logic as humans is limited.
Ali (ra) once said that 'the deen does not follow our logic, if it did, we would do masha on the bottow of our feet and not the top'.
Assuming that it is illogical to not keep dogs placing your knowledge and logic above that of the Prophet and your creator.
Philosophers have all pretty much unanimously agreed that our logic is very limited, and is very restricted, a single piece of information can upset the logical conclusions we have come to. That isn't to say that we shouldnt use logic, but we shouldnt assume that our logic is the greatest and most fail proof method by which to life.
An example is the instruction of the Prophet Muhammad (pbuh) regarding drinks. He said if a fly lands in your drink, you should submerge it completely before taking it out.
By our logic, thats completely uneccessary since 'logically' we'd be putting more bacteria in our drink than before.
However, as human knowledge developed, we discovered that a fly contains bacteria and anti bacteria on either side of its body. Thus by submerging the fly, you are negating the effects of the bacteria. So that is completely logical afterall.
Wsalam.