Sarah, I have more time now, so I'll try to add some more info you might find helpful. Thanks for adding a bit of your and Yuma's schedule - it helps to understand what's causing some of the barking, messes, etc.
Ok - where to begin? There are a couple rules of thumb to follow when housetraining a puppy, but they're pretty close to each other. One says a puppy should be let out every hour on the hour (and yes, that includes in the middle of the night, too). The other says a puppy can hold it one hour for every month of age (2 months = 2 hours). Every dog is a bit different, so there are no hard and fast rules, but puppies do need to be let out
often - and that's not just during waking hours, that's round the clock. Their bladders are very small, they can't hold it for long hours, and to expect them to is cruel. Yuma is 8 weeks old, so basically, he needs to go potty at least every 2 hours right now. A puppy his age can't be expected to hold it any longer. You put him to bed at around 10pm, and expecting him to hold it until 7am is pretty much impossible for a puppy his age. When he's barking at 4am, it's probably because his little bladder is about to burst - and he needs out NOW. Then again at 6am, he's really wanting out NOW. I know you said your sleep is valuable to you - but you signed up to get a puppy, and just as with an infant (human), they do potty round the clock, and human moms can't say hey, I need my sleep, check back with me in the morning. Puppies need to potty in the middle of the night, and we, as pet parents, must get up and take them out to potty. No way around it. He should be taken out around 1am, then again around 4am, then at 7am when you get up. I bet that would put an end to his barking. He shouldn't have to lay there in pain and hold it til you get around to getting up. Sorry - that's just the way it is.
Pertaining to the not minding outside, he minds excellently while indoors. He tries to run off while outside and no amount of "Come here, Yuma" brings him back. I have to chase him down just to have him roll over on his belly. I tell him "No. Bad boy" and he follows me back with his head hung low.
Reading that just makes me sad. You say he minds excellently while indoors. There are no distractions indoors - he's confined by walls, and altho it's fun, I'm sure he has toys, and you to play with, a house is a house. Outside is AMAZING, it's FUN, there are fabulous smells everywhere, stuff to be discovered and enjoyed - he wants to go see all of it NOW. And when you say "Come here, Yuma" and he doesn't, you have to chase him down to have him roll over on his belly? Why? First of all, I'm sure an 8 week old dog doesn't know what "come here" even means. And why must he roll over? What's in it for him when you call him? He knows he has to go back inside - and how boring is that?? How about first, you start working on the "come" command using some really good tasty treats, maybe some plain chicken - for very young puppies, you don't want to use anything that will upset their tummies, but plain chicken should be fine. Start working with him on "come" and when he does, give him a yummy treat. Repeat, repeat, repeat. Do this in the house, with no distractions - over and over and over, every single day. Then start trying it outside - but when he comes, give him a treat, and let him go again. Don't let "come' mean the fun is over - would you come? He's a baby, and he's only learning .. to learn. Give him time. Practice for a few minutes, several times a day - over and over and over. And reward him every single time. If he doesn't come to you, just keep practicing. Only call him when you
know he's going to come - set him up for success. He'll catch on.
As for telling him no, he's a bad boy *once you've caught up to him, and then he rolls over* - please please stop that. First of all, he's being a puppy - he's just trying to discover the world, and he's playing a "catch-me" game. Granted, not one you want to play. When it's time to come in, just go get him - for now, don't call him, you know chances are he's not going to come - don't call him then scold him when you get to him. Puppies go thru many fear periods, and you risk a chance of going to your puppy and scolding him, and him becoming afraid of you. I know you don't want that. I'm not sure if the puppy is automatically turning over once you get to him (in which case he is showing you he's kind of afraid), or you're actually rolling him over (alpha rolling) - in which case, please, I beg of you, don't alpha roll him. There is never a need to alpha roll a dog. Ever. You said he then comes in with his head hung low. It makes me so sad to think of a German Shepherd puppy (well, any puppy!!) to be hanging their head low - this is a puppy who should be learning to be strong, dignified, confidant, and courageous. Puppies' hearts should be happy and full of joy - an 8 wk old puppy should never have reason to have their heads hung low. I'm looking at both of mine, and altho they are both rescues and I'll never know, I almost got teary, and I pray for their sakes that no one ever made them feel that way when they were babies. I surely try never to let them feel that way now - and would never ever allow anyone to tell them they were "bad" now - for any reason.
I'm assuming you don't have a fence? For now, I'd get a long line and keep him attached - you can get long lines that are 20 or 30 feet long, and just keep him attached at all times. Also, don't leave him unattended, as you don't want him to get caught or tangled, or strangle himself. But outside, on a long line, with you, he can run and play, without the confines of a 6' leash. With a long line, he can not only get some good exercise, you can also eventually work on calling him to you from a distance, reward him, then let him go "free" again.
Try to remember that with his barking, he's trying to tell you
something. Most dogs don't just bark to hear themselves. They bark to communicate. It is our jobs to learn their barks, what it is they're trying to say. I can listen to mine bark, and know if they need something, if they're playing, if someone is approaching, if there's a problem, etc. Don't just tune Yuma out - start listening to those barks. Granted, when you first put him in his crate, if he's pottied, has a toy and/or a chew to occupy his time, and all should be well in his world, then he's barking cuz he doesn't feel like being in there, but sorry Yuma, it's bedtime (or whatever) so that's the way it is. You mentioned above, he finally quiets down, and goes to sleep. But - at 4am he starts barking. I guarantee you it's because he needs to potty. Please, don't ignore him, get up and take him out (but ideally, waaay before 4am). It's not playtime, no need to get all excited, just straight outside - go potty, (give him time) goooood boy!!!!! maybe give him a treat for going outside, then back to the crate. He'll quickly learn the routine - and most likely let you sleep a bit longer, cuz he won't be lying there suffering. After all, why should he be quiet and let you sleep when he's been desperately trying to hold it for hours. Dogs hate soiling where they sleep, so he's been so desperately trying not to for 2 solid weeks - and that's just not fair. He should be able to sleep comfortably also. If you respond to his barking (his waking you) by taking him out to potty, you're also teaching him to let you know when he needs to potty - you're telling him that "I'm listening to you, you tell me what you need, I'll try to take care of you, speak to me". He barks, you wake up, drag yourself out of bed and take him out, put him back in the crate - but he's just learned, hey, I can tell mom when I need to potty, and she'll take me out. You'll find that will start working around the house too - he'll learn to let you know when he needs to go out. It's a win/win - cuz no more messes. You want to start building a trust and a solid bond with this boy - one that will last a lifetime. Start by listening to him, start really looking in those eyes and learning what's going on in that little head. Listen to those barks, and listen to the differences in them. What does he sound like when he's playing? Demanding? Needing to potty? Afraid?
Don't tell him he's a bad boy - show him what you want, and praise the heck out of him constantly when he does right, no matter what it is. He'll start doing way more right than wrong. Ignore the "bad" stuff, praise and reward the good stuff, and he'll work all the harder to get the rewards and praise (kind of like us humans).
) will grow up with a heart so big and a love so deep for you, that he'll give you his life, if need be.
Pertaining to the not minding outside, he minds excellently while indoors. He tries to run off while outside and no amount of "Come here, Yuma" brings him back. I have to chase him down just to have him roll over on his belly. I tell him "No. Bad boy" and he follows me back with his head hung low. I hate to have to leash him while he is outside because he needs his exercise, but I fear that he will run out into the street and be squashed. 
Yes, they were different mothers. No worries, Hayley. They didn't over-breed the dogs.
). Come to think of it, I like nearly all dogs!
and then as soon as I move, off he goes. I have even tried walking off and seeing if he follows me if I get too far away, but he doesn't seem to care. 
I can handle odd looks if it keeps Yuma safe.
