Handling adding a new pet to a home that already has pets
Adding another dog to a household that already has a dog, can upset the pack order, and might cause dominance issues between the pets. A dog approaching adolescence may feel the need to test his place in the pack order. Some growling and skirmishes can be expected between dogs while they sort this out. As long as one dog does back offs or rolls over there won't be a problem, as long as the quits pushing it at that point. It should be completely bloodless, just a lot of canine 4 letter words and some pushing around while the new or maturing dog tests his position in the pack. It can sound quite scary. Sometimes the original dog will end up subordinate, sometimes not. That's not a bad thing as long as each dog accepts his place in the pack, whatever it is.
Hopefully, you're top dog and you have control over how treatment is dished out, at least while they're in your presence. If the original dog wants to be dominant and is worried about his place in the pack, continue to give the original dog his full share of attention as much as you can—at least more than the new dog. The less dominant dog can get less attention if necessary, it doesn't have to be equal, they don't necessarily expect it to be. This helps assure the dominant dog that his place is secure, and he is less likely to challenge the other dog. Things like feeding the dominant dog first, petting him first, etc., shows you respect his position in your pack and encourages others to do so too.
Making sure everyone is neutered / spayed is mandatory. Sex is a strong motivator, especially for the males. Neutering helps reduce some of this drive as it reduces some of the tension and frustration.
If you own larger or smaller dogs than your dachshund, you may have problems. Dachshunds may look at other animals as prey, especially if the other animal is small. Be it a cat, another smaller dog, or other non-dog pets (rabbits, gerbils, etc.) if given the chance, your cute little dachshund may suddenly turn into a voracious hunter. Just be careful and keep them adequately separated.
Having larger dogs doesn't necessarily mean the dachshund will back down. Size to them is not relevant. They will take on any dog of any size until that dog shows them who's boss. If the larger dog is timid, then the dachshund will feel comfortable as the new leader. If not, then you'll likely have some big-time dog fights until someone comes out the leader. It will largely depend on the nature of the larger dog.
Many years ago I had a large doberman. This was before I met my wife. He was a beautiful animal and large—115 pounds worth. My future wife had 2 long-haired dachshunds. After dating for a while we decided it was time to introduce the mixed family. What happened that afternoon wasn't pretty. My big guy was literally shaking and quaking and couldn't wait till they left. They were at him from the get go. It was embarrassing. In time they got used to each other and the doberman pretty much accepted the little dachshunds as both being top dogs.