:lol: I'll bite. I love, love, love cloth. I used disposables on DD for one weekend and hated them with a passion. I even took our cloth to Cuba for two weeks, and while handwashing diapers is not my favourite way to spend my time it was still better than carting around 2 weeks worth of leaky disposables.
So, to address your questions:
Cost: We went mostly with fitteds, specifically Mother-Ease Sandy's which cost about $12 per diaper. If I was looking to go cheap I would have gone with prefolds but as I'd never cloth diapered before I was a bit intimidated by them. In hind sight they would have been just fine at least until DD got mobile. Now, I suspect I could have diapered DD cheaper with disposables (although probably not with any of the more natural options) but the big advantage of cloth is that they can be used so often. My newborn diapers (which we used to about 4 months) have also been used by my best friend x2 and my SIL. So by the time I use them for number 2 they will have been used for 5 babies and they are still in excellent condition. Price wise, I don't think disposables can touch that.
If you were to do the same with prefolds or even flats you could diaper multiple kids for not much more than $100 I think. As Carson pointed out, there's a huge second hand market too and you can save a lot of money going that route.
Cleaning: Exclusively breastfed poop just goes straight into the machine. No need to pre-wash or pre-clean. Once DD was on solids I'd dump the poop out into the toilet but didn't get too paranoid about it.
I do think it's important to use a mild detergent without enzymes, optical brightners, scents or other nasty chemicals but rather than using a different detergent I just switched completely over to the same detergent I use on the diapers. It actually works out super cheap as the stuff I use is pretty concentrated and while it's around $30 for a bucket I've only used 2.5 buckets since DD was born - 33 months ago. Oh, and it's really important not to use fabric softeners as they reduce your diapers' absorbancy.
I'd say using public laundry is probably your biggest barrier. I generally had 18 or 24 diapers and was washing every 2 or 3 days. If I was using public laundry, I might consider going with prefolds or flats, buying about 36 of them and then you shouldn't have to wash more than once a week. In that situation I'd definitely avoid going with all-in-ones. They are a little more challenging to get really clean and you'd need to be running long wash cycles.
For transportation, I have wet bags that work awesome. I've camped with them, taken them to Cuba etc and they'd be perfect in your case as you just throw the dirty diapers in the bag, than carry the whole bag with you to the laundry, dump the diapers in the washing machine, throw the wet bag in with them then use the same bag to carry them home.
I honestly can't tell you if it would be worth the extra time, but I know some people do it and it works for them. If you're already doing laundry for a family of 4, it may not really be that much extra. You can wash them by hand and hang them to dry if you really want to but I can imagine how much appeal that probably holds.
The diapering forum here on Mothering is really awesome. I found it was a great place to find answers to all my newby questions before DD was born.
I hope that helps a little!