Hi there - goodness your post brought back memories of my ds at that age. I remember sitting up all night, watching bizarre tv at 4am and breastfeeding through a fog...
...It's so tiring, I think you are doing really well! I'll give you a few suggestions that were passed on to me - might be worth a shot.
My ds wouldn't sleep in his moses basket for ages (he was born at this time of the year too)and would scream everytime I put him down. I worked out, eventually, that the sheets were too cold and were disturbing him. I slung the undersheet over the radiator for a minute or so to warm it, before I put him in and that helped a bit.
I wasn't keen on the dummy either, but gave him one cos I was beyond caring at some point, and he rejected it, all by himself, when he was about 12 weeks old, so you won't necessarily have a 3yr old with a dummy in his mouth (which was my fear!)
I also read about a study where researchers found that for the first several weeks, babies were as likely to be awake in the dark as they were during daylight hours. At seven weeks, though, almost all the babies shifted to sleeping more at night than during the day. So you've only got a little while to go, hopefully. My friend told me to try and make day and night as different as possible, lots of sound and light during the day and dead quiet at night, but my ds was always asleep during the day, so not that helpful!
Blimey! That was long, sorry. Hang on in there - it gets better really soon.