I am not a professional trainer but have two fairly well-behaved Great Danes and have great success with some different methods for potty training, eliminating annoying puppy biting behavior, teaching sit, lay, stay...etc.
 
I've decided to post some of these in case they could come in handy. I'll have to post in installments...as I don't have time to list them all.
 
Does your dog/new puppy just not get it when it comes to housetraining?
 
1.Ask yourself if you are leaving your dog alone too long. A dog can hold it for a long time...but, not forever.
 
2.There is no exact science I have to offer here...but what has worked for my husband and I is the following for our dogs when they were relatively young.  
 
-Limit your dog's access to water a few hours before bedtime. Put them in a properly sized crate to sleep for the night. Do NOT put water and food in the crate.
 
-In the morning...before you even have time to pee run to the crate and let the dog out while trying to simultaneously attach a leash. Chances are if it's a new puppy it will start relieveing itself on the way out the door. This is expected until they learn after a few times that they are going outside to tinkle.
 
-Say the word you have designated for poopy time a few times. "Go poop" works well for me...my husband likes to scream "Drop and pop" at them now that they are fully trained. It's kind of embarrassing. When learning though, the whole family needs to agree on one phrase and use it consistently.
 
-Put your foot on the leash or wrap it around your waist and stand and ignore until the dog finally does his/her thing.
 
-Praise. Praise. Praise. Give a special treat immediately. Have one hidden in your pocket. Say, "Good poop."
 
-Some people advocate using a crate all the time...I have no real opinion. Our dogs quit using theirs after they were housetrained.
 
 
3. Here is a link with some more useful info...may work especially well on older dogs I would think...http://www.paw-rescue.org/PAW/PETTIPS/DogTip_PottyOnCommand.php
 
 
Does your new puppy bite/mouth you all the time making it painful to play with him/her?
 
1. Establish a phrase for admonishing them. Mine was "No biting"
 
2. This may sound mean, but it works. In the long run it enables you to want to spend more time with your furry friend -so I think it pays off for you and the pup.  
 
-When they bite your hands or arms or whatever in an effort to teeth and play...press their outer lip firmly but not extremely hard onto one of their teeth and simultaneously say in a firm voice, "No biting."
 
-Don't cause pain, but a tad bit of discomfort.
 
-Be consistent and do this with every occurrence.  
 
-Provide something acceptable for your dog to chew on. In an obedience class I learned that you should always give a dog a acceptable substitute if he is chewing or playing with an offlimits object. Dont' just say "nono" all the time. Tell them what they can do as well.  
 
-Don't get mad at them. They are just doing what dogs do. But, be firm and consistent. The behavior should subside in a few days to a week. Don't think you can do this every now and then...you must do it everytime. It works on the principles by BF Skinner alot of us probably learned in school-operant conditioning. If the dog learns 'when I bite, I feel discomfort' it will soon stop biting. The behavior will be what is referred to in the theory as 'extinguished.' More on this in the next question.
 
Is it you or your dog's fault?
 
Our eldest Dane, Josie, has learned to bark and bark and bark if she wants to come inside. It is annoying. We have tried ignoring her...but other times we've let her in when she barks. This is not good. It only makes the behavior of barking stronger.  
 
A long time ago BF Skinner http://evolution.massey.ac.nz/assign2/PH/skinnerwebpage.html made a Skinner box putting a rat in it. When the rat pushed the lever it got food. After a while he quit giving the rat food and so the rat quit pushing the lever (after pushing it feverishly many times--behavior gets stronger before extinction). Now...what if he gave the rat a pellet on average every 5-9 pushes of the lever? Well, the rat learns to push the lever faster so he can get a pellet. This is what happens when we sometimes let Josie indoors when she barks...she learns to bark up a storm because we will give in.
 
I suppose the answer is to leave her outdoors and let her in when she is not barking. To show her we decide when she comes in...not her. However, I'm freelancing from home now and can't have a dog barking in the background when client's call -so I guess Josie will win this battle.
 
The whole point of this question is to get you to think about consistency. It is key!!! I admit, we aren't that good at it in some respects.
 

Mine rough house some...but not tons. My oldest one likes to gallop at top speed across the back yard which is scarry. We all just freeze and trust her to dodge us as she goes by. One time she missed and broke my husband's finger. The other fatter one (Abbey) doesn't run at top speed, but she does get wound up when you take out a spray bottle or if someone plays the harmonica. She gets all playful and fraps around. I just learned the word frap a few years ago. It's when your animal darts this way and that totally out of control for a few gleeful moments. It's always funny to see.
 
Abbey mostly enjoys playing with the little dogs whose heads she can get in her mouth. She does it in a friend way....they are soaking wet when she is done.
 
Are your female or male or both?