What do the suicide rates have to do with gun control or crime?
UK gun crime rates are still among the lowest in the world.
Yes, as far as I can tell, gun crime rates did increase after the ban on handguns was enacted but is now on a downward trend.
Could this be because the existing illegal handguns are still being secured?
Maybe because the legislative powers given to the cops have finally started to take effect?
The question remains though...If having firearms freely available is so effective in preventing gun crime then why is the USA so high in gun-related crimes in comparison to the rest of the world?
No, the two are not related. Gun-related crimes are very rarely perpetrated on people who themselves have guns (gang violence notwithstanding), so the level of gun-related violence in the US is not a reflection on how effective gun ownership is at preventing it, but rather a reflection on how bad things like "gun-free zones" are at doing so.
If you wish to compare the US to other countries in terms of crime, you're going to have to look at other factors to figure out the difference. Which factors would depend on which country you wish to compare us to, since believe it or not, not all countries are alike and interchangeable.
Consider England for a moment. Gun control advocates love to point out that England, with its strict gun control laws, has a lower homicide rate than the United States. While true, one also needs to consider that even before the implementation of those laws, England had a lower homicide rate than we did.
Furthermore, the United States has a NON-gun murder rate that is higher than all of Europe's TOTAL murder rate. So it would seem that guns are not the issue.
I saw Japan mentioned earlier. While it is true that Japan has lower homicide rates, it is ALSO true that Japanese-Americans have lower homicide rates than the rest of the country, despite living under the same laws and having the same access to guns. It would seem, then, that we should be looking at cultural factors, rather than guns.
If you truly want to understand why New Zealand is different from the United States in regards to crime, you must consider the question: In what ways (not just gun-control laws) is New Zealand different from the United States?