I've always found the less sleep mine get, the worse they sleep. Children have different body clocks to ours and can get very unsettled if we do not adhere to them in my opinion.
I think at 6 months they're old enough to do a bit of gentle sleep training with. I used the baby whisperer pick up put down method (PUPD). So when they were due for a nap, I would start an identical routine everytime so they knew it was nap time by my actions/ words. Then I would lie them down after and when they cried, I would pick them up remaining right by the cot all the time, once they stopped crying, I'd lie them down again. When they started to cry, I'd pick them up, when they stopped I'd lie them down. Over and over again, never leaving room. The first time I had to pick them up 70 times, the next nap time, 50 times, the next sleep time 30 times etc etc. It took 3 days. This worked for my daughter brilliantly. They get a bit cross the first time you do this but they soon get the message that mummy means business.
However with my son that method didn't work so well. I did the same usual method for nap time and then would leave room for 5 minutes and then go in pick up until calm. Lay down, leave again wait another 5 minutes and go in etc. Until he passed out. That worked far better with him after a few days. But you have to do what you feel comfortable with here.
Sleep routine wise, I woke mine at 7am as otherwise the rest of the day went to pot, put down again at 10.30am for a nap. Then kept awake until about 2 or 3pm, then put them down for another nap, and then put them to bed after a bath at 7pm ish. This worked well for my two and still does with my 14 month old. daughter at 2 and a half still naps at 1pm for 2 hours as well. If my son misses a nap he sleeps worse at night and is harder to settle as he is so overtired.
Both my children already had 2 teeth by 6 months, so if they woke in the night and weren't due a feed, I'd put some bonjela teething gel on their gums and leave again and see if they'd go back to sleep. Often they did. If they really wouldn't settle still, I did offer a feed and they'd sleep afterwards for 3 hours until the teething gel wore off again. It took with both my kids until 9 months to sleep at night without waking for a feed at 3am. I kept the 11pm dream feed until they weren't interested though and didn't take much.
Every baby is different but that's what worked for me. I love this book but didn't use his sleep training advice but everything else I found helpful.
Whenever mine were ill I would have to roll with it and re-train them once better however.
If they're waking from loosing the dummy, it maybe a good time to go cold turkey if sleep training anyway? So they learn to self soothe. I found sleep training very hard and cried and felt guilty but looking back I never regret it now.