I had just the same starting trouble on my Rover-engined 1999 +4.
After a very frustrating period of checking or changing various components it turned out to be the starter motor in the end, which was drawing too many amps apparently. Morgan had a batch of poor starter motors fitted at about this time, so it is a known problem. I changed mine for a 'High Torque' PowerLite unit from Eurolec Components at Brierley Hill near Dudley in the UK(at a cost of £130 including the mounting adaptor)and since then it's been no trouble at all and spins over a treat.
As for your other problem, you will need specialist diagnostic equipment that may prove hard to find, as this Rover system seems to be now obsolete. I eventually found someone near to me in Leicester who had all the correct (and very expensive) kit, who is a specialist in fuel injection systems, so it might be worth trying a fuel injection specialist local to you.
It will certainly be worth checking all the earth fixings on the car anyway, just to make sure they're OK.
For peace of mind there are also specialist ECU testing companies that can check the ECU off-car for you for about £35.
Good luck - and welcome to the joys of Morgan ownership!