From the limited information available there would appear to be three issues.
1. Knowledge and experience of the firearm requested. This falls into the category of 'public safety'.
2. Motoring offence convictions leading to disqualification. This can be interpreted as having a disregard for the Law and/or being irresponsible. It is all a matter of fact and degree as to how much weight is placed upon this aspect. It is a 'fit to be entrusted' issue.
3. The current situation with the mental health and wellbeing of the applicant. This will be assessed according to the evidence supplied by medical professionals and your GP should be asked for a factual statement.
In reverse order and noting the adverse life events experienced by the OP, it is unsurprising that it would have an adverse effect on morale and wellbeing. If the depression is purely reactive and responding to treatment, it should not alone prevent the issue of a certificate. The licensing manager will consider whether the current state of mind/morale presents a risk to public safety (including risk to the applicant), should a certificate be granted. This is something that is highly personal and sensitive to each individual case.
In relation to the motoring offences, the number and duration of offences, their seriousness and whether persistent will be taken into account. For example three speeding offences collected over a period of time would be on the lower end of the scale whilst driving without insurance, dangerous driving, drink driving would be viewed more seriously. A person with two drink drive convictions within five years would be unlikely to be granted a certificate and if a certificate holder would be revoked.
Knowledge and experience is more nebulous and sound advice has already been given by others, regarding obtaining written references from those with whom you have used a similar firearm. Level 1 DSC is also viewed as showing commitment.
Having said all this, it may be that the most sensible and pragmatic decision could be to postpone the application until some of the issues have been resolved.
That would avoid forcing a formal refusal. However, should you chose to withdraw/postpone the application, I would reccomend that you politely ask the firearms licensing manager to inform you in writing as to what you need to achieve, in each aspect before favourable consideration would be given to granting a certificate. You will then know where the winning post is and how high the hurdles are on the way.
Some may recommend a more robust and combatative approach, but in view of the range of issues outlined, a more conciliatory dialogue may bear fruit before too long.
With best wishes. U Nm.