Weird Starting Issue

We have a strange starting issue with our Nissan Micra SE K12 2006.

It first happened in March when it was freezing cold – it would turn over, battery was strong and kept going and going, the starter was trying its best but the car just wouldn’t start.

Got a mechanic in and he got it started (eventually), but he didn’t really know why it started or what he did to get it going.

Ran fine again for 3 months and in June it did it again. Took it to a garage, they couldn’t start it, took it to another last week and it started first time and has done every day since! So they don’t know why either.

Before giving up completely it gets slowly worse over a week or two where it won’t start, but then we take the key out, rotate it and try it the other way and it starts. Then sometimes that won’t work and we try the second key and that works. Then it gives up entirely.

Does anyone have any ideas at all what it could be?

We’ve so far discounted:
Timing chain – the latest garage have ruled out this as an issue.
Top crankshaft sensor – replaced this in June with no change.
Spark plugs – new in April and made no difference.
Coolant temp sensor – replaced after engine warning light came in April.
Battery – turns over and over so seems OK, and it’s starting again now.
Starter – turns over and over so seems OK, and it’s starting again now.

Nissan replaced an ECU relay as a recall in May, but this was after the car failed first time.

Update: Got it back from the garage. As soon as it came off the flatbed, they said it started and started every day for the week that it was sitting there. We took it home and within a week it refused to start again…

Today I jumped it with my car and it started immediately. Is it possible that the battery is faulty somehow? The car lights up, everything comes on, it turns over (keeps going and going), the immobiliser light goes out when I put the key in, and when it does start it runs perfectly.

One last thing…if I run it down our hill and try to bump-start it, it doesn’t work. Just grinds to halt.