First, unputdownableness seems to be not unusual, but also not normal. No idea on the figures, but might guess would that about 10-15% babies are unputdownable. It's not unreasonable to be surprised - none of the baby books I read while I was pregnant and neither the NHS or NCT antenatal classes I went to mentioned that some babies can't be put down! The only book I've come across which discusses babies like this is The Baby Book by Sears.
It's hard to know what has what effect, but what I think may have helped with our baby is to start by choosing a place that we'd like to put him down (bouncy chair is probably a good starting point as it's safe and lots of young babies seem to prefer being vertical to horizontal). Then when he's in a happy, smiley mood and not tired and hungry, put them in it for a short while and give them your full attention singing, talking to them etc. Take them out before they get grumpy if you can. Repeat and gradually try and increase the length of time and possibly playing with toys with them at the same time. Then start leaving them for very short periods of time (I always say the same words when I leave our baby, no idea if it helps!) staying within sight still, and gradually increase the length of time you leave them for and eventually also going out of sight.
I find our baby is happier the better he can see me, so if you're doing a chore like ironing, I'd recommend making sure they are well-positioned and that you go back and give them attention every now and again.
Don't expect miracles - you're unlikely to get your baby e.g. falling asleep but you might get the odd 20 minutes or so several times a day when you can rush around doing all the things that you have saved up that you can't do wearing a sling.
In terms of tummy time, apart from on your chest etc, if you breastfeed, look up biological nurturing. I've found the sling has also been very good for our baby's neck muscles. Our baby's physical development certainly on a par with other putdownable babies his age and he does very good mini-pushups!
Good luck, and would be interested in any tips from anybody else with unputdownable babies!