Hi Jason -
I just read through this entire thread and I'd like to add my thoughts on it.
First of all, I can relate to some of what you're going through. I took on a 5 month old shepherd several years back who was the most frightened pup I'd ever seen. I don't know how many times I thought "OMG, what a huge mistake this was!". She was afraid of everything. If I raised my voice, she cringed. She pooped and peed in her pen and then would spin nervously, spreading the poop everywhere. I didn't enjoy her at first. But I made a commitment and it was my responsibility to deal with it one way or the other.
These are the things I'd recommend to you regarding your pups.
The most important part of training is to build a relationship with your pups. This has to be done one-on-one and it has to be done in a calm and happy manner. Using the higher silly voice is a good step (as you're already seeing). Show your pups that you are the one to come to when they want companionship.
First - let them be puppies and don't stress about the training. If the clicker is frustrating you, set it aside. You don't need it in order to housebreak or even leash train. Spend the majority of your time enjoying these pups, but do most of it separately. Play with them. Explore their personalities and find out what motivates them. You NEED to know this in order for positive training to work. Try a variety of treats: chicken breast lightly dusted with garlic, microwaved and cut into tiny pieces; beef heart prepared the same way; vienna sausages; liverwurst; tuna fish; mozzerella cheese in tiny pieces, etc. Find out what they like to do and consider using that for a motivator. For instance, if they really like to go out the door, then you may use opening the door as a reward for walking to the door on-leash. Every dog is motivated by something and you need to figure out what it is.
Second - the pups do need to be separated and that means inside and outside of the house. This doesn't mean they can't get together and play some everyday, but they shouldn't sleep together, walk together, potty together, etc. The problem with multiple puppies is that they have little need for a human if their sibling is there with them. You will create the need/desire to be with you by separating them and then being there for each puppy.
Third - housebreaking - no freedom in the house unless the puppy has JUST pottied outside. Use crates, ex-pens, long lines, etc. Use a long line outside. If the pup gets to the end of the long line and throws a fit, just stand there quietly and watch him. Don't give in to it. Eventually the pup will move to a place where he isn't pulling against the leash and the tantrum will end. I wouldn't use any sort of conciliatory voice or words while he's throwing the fit - this can easily be construed as reinforcement of the trantrum by the puppy.
IF you absolutely don't want to use the long line, then get a pen. If not a real chainlink style pen, buy an exercise pen and set it up outside. Take the pup directly to the pen and put him in to potty. If he doesn't potty in a reasonable amount of time, take him back inside and put him back into a confined area. Don't allow him to run into the house - this should all be controlled but controlled in a calm and quiet way. If the puppy potties in the house, it's 100% your fault.
Fourth - leash training - once you find out what motivates your dog, use it as a reinforcement in leash training. This works best if you've found a good treat your puppy likes (by the way, don't free-feed .. if you are, you will have a lot more trouble in training with treats). Take the really special treat you've found the pup likes, put the leash on the pup and then stick that treat right at your puppy's nose and back up a step or two while praising constantly. Let the pup have little nibbles of the treat. Then lift your hand a bit so the pup can't quite reach it and back up another step or two, praising like crazy as the pup follows - and give it another nibble of the treat. Back around your living room or hallway with the pup following in front of you and reward frequently. The first step to leash training is to teach the pup to follow you.
If the pup goes to the end of the leash, stand there and be quiet. Let him throw a fit. Don't react, don't reassure, don't scold, just let him figure out that fighting the leash isn't going to work. Once he stops fighting, step forward a bit to give him some slack and encourage him to come to you (even if you have to lure with the treat) and then reward him when he gets to you.
Not using the leash because the puppies don't like it is never going to teach them how to handle it. When the puppy fights the leash, don't think of it as a bad thing - observe, learn, let the pup work through it and simply be an observer UNTIL the pup offers a behavior you like (calming down, moving toward you, etc.). And then praise and reward!
Fifth - car rides - this needs to be done gradually and DAILY. Start with putting each pup (individually) into a crate in the car and then feeding them a yummy treat. Then take them out. Do this a couple of times a day for a few days at least. Then put them in, shut the car door and start the car. Don't go anywhere. Turn the car off and let the pup out (but wait for the pup to be quiet and calm if he's fussing). Do this a couple of times a day for a few days. Then drive down the driveway only. Again, for a few days. Gradually lengthen the trips.
It may take a long time before they can travel in a car without fear behaviors and without vomiting. If you've ever been car-sick, you'll know how miserable it is. As far as their fussing and being upset, I'm sure that having two of them makes it worse because they encourage each other. One gets upset, the other does too and it just gets worse and worse. By separating them and working on the car rides one at a time you will help avoid the problem of having them react to each other.
I've been teaching obedience classes for a LONG time and I just don't recommend to people to get two puppies at once. It is twice the work and often ends up with major problems because puppies naturally bond to each other. It takes a huge amount of commitment in order to make it work. I hope that you have the commitment - if you feel this is going to be too much for you, it is much more fair to the pups to rehome one now and not after it's already developed problems. It would be good for you to seriously look at your feelings on this.
Re-read otch1's recommendations .. she makes a lot of sense.
By the way, you'll get a lot of suggestions and training advice from a large variety of people. Some of the advice contradicts other advice. You can't try to do everything. Even the books you're going to read will contradict each other at times. You need to really settle into a course of action and follow that consistently and patiently in order to see results. If you try one thing for a couple of days and switch because you're not seeing quick results, you will just screw up those pups. Often it takes weeks to get consistent results. So please read through the recommendations and decide what you're going to do and then stick with it for awhile.
Good luck and I hope that things get better soon -
Melanie and the gang in Alaska
By the way, the light colored shepherd in my signature is the dog I talked about at the beginning of my post. It took a long time and a LOT of patience and persistence on my part, but she turned out to be a highly bonded, loving girl who was very responsive and trusting.