Well, I don't think the Bible says animals don't have souls, per se, though it's often interpreted that way. What it clearly says it that humans reflect God in a special way and are made higher than other animals.
I don't think this verse in Timothy is saying anything about the household, but it does say/imply that men should govern the church, and teach therein. Why might this be? Well, one can speculate, of course. The Bible certainly teaches a more complicated view of men and women and their roles than the most popular modern view of near-complete fungibility. I don't at all see it as misogyny -- mistreatment of women in the ways usually associated with misogyny is violently discouraged in the NT -- but you can, of course.
"How can one have faith in the inherency of scripture, or in God when he insists on things like this. (The second even being from the New Testament.)"
Well, I think it's true that if you're going to believe in inerrancy, you can't pick and choose. You can interpret, of course, and should be careful to interpret carefully using all means available. But if you conclude, as here, that God is saying women should not teach in church, then it doesn't make much sense, to me, to say that you believe the Bible is of God but that it's just wrong here. What intuitions would we have that would be more reliable than God?
So to decide to accept the Bible as God's Word is to commit to accept such things. That doesn't mean you have to just quell whatever in you finds them not to make sense or to be wrong-seeming. It means you commit to work through and get to the bottom of why they do make sense, and are not wrong, after all (or else, in some cases, why you maybe were misinterpreting them).