Firstly, in response to your question, I have obtained marriage licences for post 1837 from the relevant Register Offices, In Michael C's case it was Durham - usual charge is around £9-£10. I only once got one from Ancestry & the cost was extortionate. Never again. Prior to 1837 I am using parish transcripts or, if obtainable as sometimes on LDS, I use images of the original or Bishop's transcripts.
I've contacted Bernadette Mills at http://trees.ancestry.co.uk/tree/7788426/person/726917575http://trees.ancestry.co.uk/tree/7788426/person/726917575 as you suggested. She had no additional info on Michael BUT she was able to confirm that the marriage cert for John C & Bridget confirms John's father as James.
Following up on the possible Patrick Blewitt link I found a record of his marriage to Ann Convey in Jul-Aug-Sep 1853 in Newcastle upon Tyne. From the ages given on the 1861 census Pat would have been only 15 at the time and Ann 20. The census also shows them both as being born in Ireland.
So the family could be; Ann abt 1833 Michael abt 1835 John abt 1842 ...only I can't see a gap of 7 years between Michael & John so it's likely there were other siblings, possibly another 4 or 5.
It really does seem likely that the whole family came over, as the Coyle's did, so perhaps I need to see if I can find anything along those lines.