I am also still struggling with my 5 year old. I don’t want him to follow me to the basement because we have a lot of tools down there that could hurt him. I still have to carry/drag him back upstairs. I don’t know what else to do.

Thanks for the ideas. She knows how to work the gates we have lol. I will try reward idea. Maybe a better way to describe-I will go outside on the porch, she will open the door to try and follow. I will tell her, No, go back inside, she will simply continue on her path to come outside no matter how many times I repeat it. I have to physically take her back inside. My wife says we shouldn't spank her, but I am limited on skillset :)

You could try taking away something away that’s important to her. You could tell her “if you poke at the puppy again, you won’t get your teddy bear for the rest of the day”. Stupid question: Have you shown her how to nicely pet the dog, and show her hand over hand the motion? The most important thing with the dog in particular is safety. If she’s actually hurting the dog, that definitely needs addressed, but if it’s not hurting the dog, and she(?) doesn’t yelp or try to get away, the dog just probably just sees it as attention, and there’s no reason to intervene.
I am still struggling with my now 5 year old with listening. I’m STILL doing hand over hand with picking stuff up, and he is upset the whole time. I’ve tied a knot at the end of my rope, and am hanging on. Rewards might also be beneficial. If she listens to a request, she gets a sticker, or a special cookie, whatever you can make consistent. She may need the reward for a while, but she might do it for that special thing. Another trick (if she’s able to) look her in the eyes when you give her a command so you know she’s paying attention. Most kids have selective hearing and some kids need a bit more direct of an approach. Always keep it simple. If you’re telling her not to enter a room, just say that. Don’t try to explain why, because your main command could get lost in all the other words. In your home, you could also get gates. They make gates that open so it’s easier to walk through.

just used that as a basic example. The biggest problem is her going to poke and prod our new puppy and we tell her no and she does it anyway.

Are there some natural consequences you can employ? It probably depends on why she can't go in the room. Is it a safety issue or is it someone else's personal space or etc?