Hi!
Sorry to hear you are having a hard time... I can totally sympathise, being in a similar situation to you - DS2 is 6 months and has 2 older siblings. Trying to fit naps and feeds around school/nursery runs is so hard isn't it! I'm sure someone will be able to give you great advice about establishing EASY with a 5.5mo etc but just wanted to let you now how I (sort of!) manage with mine in case it helps you.
DS2, at 6 months, has just about reached the point where we can get home from Nursery/School run in the morning and put him to bed without him being totally OT and/or falling asleep on the school run which makes life SO much easier. At 5.5 months I would have thought it wont be too long until your LO can mange 3hrs awake time first thing which might mean you can get home from the nursery run and get him into bed for a nap without him being too OT. I expect that will make things much easier for you, as it has for us. As for encouraging independent sleep, I worked on this with my LO when he was younger (about 8 weeks I think) but what I did was concentrate on getting him to sleep in his bed using shh/pat for one nap a day when my other 2 children were out at nursery and school (that way it didn't matter if I was up in his room for ages shh/patting) and for the times when they were both around, I'd AP naps in the sling/car seat/pram etc.. The time it took for him to go to sleep using shh/pat very quickly decreased so that when I only needed to shh/pat for 5 minutes or so, I then was able to help him nap when the other two were also at home as I felt able to leave them downstairs for that amount of time. Then before long he didn't need any shh/pat which was amazing! And as for short naps, I'm sure someone can offer you better advice than me... But we had 30 or 40 minute naps only up until 4 months or so and still get them now in the afternoons especially. Getting a good first nap though in his bed seems to work well and it doesn't matter too much if his other two naps are 30 or 40 minutes and out and about in the car seat etc..
Good luck and I hope things get easier for you soon.