It can make a difference. Someone with suicidal thoughts who has a gun, the gun can give them a quick and easy method of acting on it, with high certainty of success. Without a gun, they’d have to resort to other methods, which involve more work, have lower likelihood of success, and greater risk (and hence anxiety) of the “you don’t actually die you just end up severely disabled” outcome

For someone dead set on killing themselves, it doesn’t make a big difference. For someone who has a sudden impulsive thought of suicide, having a gun can mean they die, without one by the time they’ve worked out how to kill themselves the impulse has passed

There isn’t a simple correlation between gun ownership rates and suicide because suicide rates are determined by many factors of which gun availability is just one. But I think it is a real factor, and there is research which supports that it is, e.g. https://pubmed.ncbi.nlm.nih.gov/36652694/

No but it increases the risk of a suicide attempt and it's lethality. The link between homicide and guns is small and nebulous, the link between guns and suicide is clear and much stronger.