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How much DNA does he share with the first match that you think is paternal, but Ancestry is assigning them as maternal?

I justified the upgrade as a Christmas present to myself! 

And I guess it's a similar situation for the North American match - confirmed line back to a man born in North America around 1810s.  Father's name not recorded anywhere they could find, and on a census the son had said his father was born in Ireland. 

With the other match there wasn't any particular burning question.  Turns out some of the Y37 GD=0's relatives are autosomal matches to some family members.  Not many, and it was sheer luck that some people were at Gedmatch where it is easier to work with the info.  This family had thought their earliest known ancestor had a Scottish forebear, but he must have been from the West Country. 

The results are presented merely as a list of names (no handles or nick names here) and they in theory should correspond to your own paternal name.  Like I say, about half of mine are variations of my family name.  It should be clear what name you belong to and there are named discussion groups you can join.  You get all sorts of other haplo information, maps and the like but its all quite generic and certainly of no genealogical use, which I knew when I took the taste. That was the origin of my question - what would my $300 specific haplo group really tell me!!  ftDNA is awful at answering that simple question

Upgrading to Y700 would provide a better idea of when branches occurred. 

I funded a test for one of my male relatives whose patrilineal lines leads back to Scotland.  We're fortunate with this line that we have records back to the 1600s because they'd been granted land.  According to these records the male line has been on the brink of extinction since the early 1700s when four sons were born.  One brother died young before having children and two others had a daughter.  The remaining son married and had several sons but two didn't have children and the other had daughters, leaving just the one son who was in his 40s and unmarried.  He finally married and was the gtgt-grandfather of my tester.  So, if the tree and records we have are correct anyone else matching on the line should be related from prior to 1700.

At the Y37 level there were 5 matches, 4 of whom with the same surname as the tester (just different spelling really).  I was pretty excited about this so within a week purchased a Y700 upgrade, and a few months later one of the matches  (in North America) were interested in doing the same, and one in Scandinavia (who descends from a soldier from Britain)

Dating is far more accurate with Y700 than Y37, and it shows the date of MRCA for the GD=2 match at Y37 is around 1650ish (give or take quite a lot) and for the GD=4 (different surname) it's around 1450.

This isn't everyone's cup of tea, but it's been useful for the North Americans for pinpointing where their origins lie as they could only get back as far as a census record saying the ancestor was from Ireland.  The one in Scandinavia has probably benefitted the most even if the connection is 600 years ago.  My tester has gained less as we already had good information, but it has confirmed there has been no hanky panky up our line for 100s of years.  I'm glad I paid for the upgrade, as it encouraged two of the closest matches to do the same.  I have no doubt more matches will pop up who will further fill in the branching.

The other tester from another line paid for the test themselves, and we've found it very useful as there's a match whose MRCA is probably about 6 generations back (mid 1700s ish).  Unfortunately he died several years ago (before my relative tested) but we've been able to connect with other members of the family and collaborate on our research.   They're an exact Y37 match.  Some other Y37s have the same surname - 2xGD=1 and GD=2.  The GD=1 are from the same location as the tester's gt-grandfather.  But most of the Y37 matches aren't very useful as Y37 doesn't separate the closer matches from the more distance ones as well as the higher level tests, which may explain your lack of enthusiasm for Y-DNA testing.   

The mutual matches should give you an indication of whether the connection is up another line.

How many segments?

Sorry - the two second cousins had a mutual GREAT-grandfather William, not grandfather.