Im so sorry you are having to go through this right now. I really wish all moms coudl have an easy time nursing their babies. You deserve a calm enjoyable babymoon, and I hope you get some good support with this.
First of all, you need to see a LC or midwife who is experienced with nursing problems. The swelling and pain (apart from the expected engorgement at first) is not normal. If it is mastitis, not nursing will make it worse. Your breast needs to be emptied one way or another. Maybe try olive oil instead of lansinol and make sure your bra is well fitting. Anything tight on your breast can cause mastitis. Also, going braless may help you out. Keep your breast dry and clean so it can heal. I have known many mother's have painful, cracked, bleeding nipples and still continue to nurse, including myself. Its hard and it hurts, but it can get better.
For the nonstop nursing, have you tried bedsharing? Especially in those hard hours early in the morning. It seems your LO just wants to be at your breast with you. They do that, its normal. I havent had a baby that liked to be taken off the breast and put in a bed alone. And for me its not wroth the fight to try and get them to sleep alone when all they want is to be next to me nursing. I also agree with nursing on one breast at a time, even the same breast for two or three feedings in a row and pumping the other when needed. My infants nursed almost nonstop for the first few weeks. I always thought that was normal, it is for mine so I dont try to do any different. I see that some parents (Im not talking about you, just people I personally know) seem to think babies can go an hour or two without nursing. Their stomachs are the size of their fist, thats small. They need to be fed often. But they also nurse for comfort, its an instinct to survive. They just need to be at your breast. It will slow down, but now istn the right time. Just give it a few weeks and you will see, he wont need the breast so much.
As for the throwing his head back, have you tried other positions? He may feel scared if you are trying to force him to your breast. Have you tried laying on your back and letting him lay on top of your stomach? That way he isnt being forced to your breast, he can find it himself. Or even try side lying. A few pillows behind your back and head can make this a very comfy position for sleeping too. Im not a fan of swaddling, but maybe wrap his arms in a blanket so they cant get out?
As for the formula, thats your call. Your baby is very small to be given both the breast and bottle, it may cause more problems than it solves. I dont know how hard this is for you, and what kind of support you have. And I dont know how important breastfeeding is to you personally. But formula feeding has never sounded like an easy solution. I see getting up all day and night to make a bottle, often with a screaming baby, heating it, washing them, and having to keep buying to be such a huge burden. Not to mention the possible health problems it can cause.
If you have other questions, I know there are a lot of mom's here who would do anythign they can to help you out. I hope things get easier for you.