I'm sorry to hear that you're going through a rough time. The good news is, you can get this under control while they are still young.
I think once you get the sleep issue taken care of, the days will be much easier for all of you. Do they have a quiet, consistent, and calm bed time routine? If not, this would be very important for you to establish with all three of your kids. It might look like:
1. everyone help clean up
2. bath time and lotion
3. brush teeth
4. stories
5. hugs and kisses
Something that will help wind them down and let them know it's bed time is essential. Also, it is important that their bed time is early enough. Staying awake too late at night can cause their bodies to release hormones that will keep them awake. For kids under three, I would think that they should be asleep no later than 8. It may seem contradictory, but kids usually sleep better when they are fully rested, not when they are over-tired.
As for the waking up, establish a rule for yourself when they wake up. Whatever it is, be consistent with it and do it every night. I would suggest just not going in to get your kids when they wake up. They may not like this but they will get the picture after a few nights. None of them are at the age where they need to be fed during the night, so there's really no reason why you should have to be up with them. This takes several sleepless nights to get through at first, but it will be worth it when they all start sleeping through the night. If you do need to go in the room, don't talk to them or stimulate them, but just gently pat their back or comfort them without getting them out of bed. If that doesn't work, you may need to stay out of the room all together.
Once you and your kids are getting better sleep, you will be more able to handle the behavior during the day. But, whether you're tired or not, you need to be consistent with the tantrums and the way you handle behavior. NO exceptions! You are right, bad behavior does not mean you get a cookie!
As you know, kids this age feel very strongly about what they want and have very strong emotions. Helping them learn how to effectively deal with these emotions can help. Kids don't automatically know how to go from raging emotions to a calm state. This is a skill that parents have to teach.
--Sometimes they need to get out their anger in a tantrum. Just ignore it and make sure they are safe. When they are a little calmer, calmly talk with them about their feelings: "I see you're very sad/angry right now. I can't let you have... but you CAN have..." Validating their feelings is important. They need to know that it is OK to be upset, but it is not OK to scream and throw a fit. By naming their feelings, it can be less confusing and overwhelming to them. It helps them sort out their feelings.
--Give them an alternative to screaming or whatever the bad behavior is. "You may not throw toys, but you can throw a ball." or "When you are angry, you can use your words to tell me that you are mad. You can say 'That makes me mad' or I don't like that' "
--Stick to your word when you say something is not allowed, but focus on what they CAN do or CAN have instead of what they can't. "I'm sorry you're sad. We can't have candy at the store but we can have a snack when we get in the car. Would you like granola bars or fruit?" Giving them two acceptable choices throughout the day can also let them feel like they have some control in their world, but not an overwhelming amount.
--Also, make sure your tone of voice is firm, but respectful. If you sound worn out, overwhelmed, and winey, your kids will see that and might try to be more persistent, or they may wine more themselves based on your mood.
The most important aspect is to be consistent. Just as important though, I think you need to remember to be silly and have fun with your kids too. I'm sure you already do this, but sometimes when we're exhausted and frustrated (not to mention the marital stress you're going through) it can be hard to remember to be positive with your kids. Letting loose and being silly with them can remind them how much you enjoy being with them and how important they are to you. It can bring you guys closer together and relieve some of the stress of disciplining. When they are always being redirected or disciplined, it kinda starts to all sound the same and they stop paying attention. Make sure you lighten the mood sometimes and you will probably see some improvement in the behavior.
Sorry this is so long! I work with parents and kids under three so let me know if you would like any additional help or ideas that are specific to your situation.
Best wishes! C.