It's good to see a few people that actually understand where I'm coming from, thanks! I just think it's funny that people will argue so passionately that you are a bad DM when all it comes down to is a difference in play styles or preferences.
Oryan, I'm pretty sure I know where you're coming from since I DM'ed in a similar style for many years. I'm not trying to paint you as a bad DM, I'm sharing my experiences. My campaigns have only gotten better the more open I've been to to player input/concepts, even when they're not exactly to my tastes.
If players have ideas for the campaign, I am not against listening to them and considering them. I have used lots of player feedback and implement ideas from it into the campaign.
Great!
I grew up playing D&D with the understanding that my PC has no control over the world around him other than what he does in game.
I think most of us did. But that understanding is neither right nor wrong, and I can tell you from personal experience I've had great results in recent campaigns by making the players 'partners in the creation'; giving them narrative authority beyond control of their PC's actions.
People are accusing me of being a controlling DM, but isn't it just as controlling for a player to insist that my gods in my world do what controlling players want them to do?
No. The DM controls most aspects of the campaign setting, all a player has is their actions and their background (and whatever additional narrative control the DM doles out --which is a good idea, BTW

).
A player that is calling me controlling for simply trying to "control" my campaign world sounds a bit hypocritical to me.
That's silly. In a traditional D&D campaign, the players don't 'control' enough to be controlling.
If a player wants to run my world, then why don't they just DM their own world?
Why not share yours? You
did invite other people to play in it.
They can name their pets whatever they want as long as it fits the tone of our campaign.
So players are free to name their PC's something you like. Doesn't the DM get to name enough? All the people, places, and things that aren't the PC's (and their pets). Practically speaking, doesn't the DM have enough tools at his disposal to maintain campaign tone without needing to control PC naming?
Heck, I thought I was actually being a good DM by not saying anything about how the first pet name bothered me.
You were!
She blasted me for allowing the 1st silly pet name to begin with because I "got her hopes up". So next time I won't be so lenient I guess.
It's not lenient, it's being respectful and accommodating to people w/different interests. Those are good things.
That is a very hard question to answer. All I can say is that I have an idea for the tone I'm trying to set.
It
is a hard question, but one a DM should keep asking themselves. "How can I balance my likes with those of my players?" All I can say is that I've come to see player input that I don't immediately like --be it a character concept, name, potential story arc--as a DM'ing
challenge, simply acting as arbiter of campaign taste/tone/flavor seems boring to me. Besides, like I keep saying, the DM has control of the most of the 'engines of tone production', I have trouble seeing how some incongruous tonal elements attached to the PC's can seriously challenge that.
And I've found the best way to respond is to simply 'say yes' to players --unless they're only after gross mechanical advantages-- and then spin things from there. Acknowledge their concept, then create challenges from it. No one likes being edited, especially during their leisure time.