Answers:
1. Any conviction before a criminal court (including, for example, motoring offences) creates a criminal record; the penalty imposed by that court is irrelevant.
2(a) Applications for shotgun licences are exempt from he provisions of shotgun certificate applications and all criminal convictions MUST be declared. (It's a criminal offence, possibly leading to a custodial sentence, to fail to do so).
From the notes accompanying the application form:
"You must not withhold information about any conviction. This includes motoring offences, binding overs, formal written warnings, cautions and convictions in and outside Great Britain, and (by virtue of the Rehabilitation of Offenders Act 1974 (Exceptions) Order 1975) convictions which are spent under the 1974 Act.
A conditional discharge and an absolute discharge both count as convictions for this purpose. Details of parking offences and fixed penalty notices do not need to be declared."
https://www.gov.uk/government/uploads/system/uploads/attachment_data/file/352534/NotesForm201A2014.pdf
2(b) Some offences automatically bar a person from holding a shotgun certificate. However the one you refer to does not. The decision whether to issue a certificate will be a discretionary one, with the period of time that has passed since the conviction likely to be taken into consideration.