Universal BG checks are a no brainier, only the most hard core second amendment absolutists agree that UBGs are a good thing. The chief argument against them is enforce-ability but most people are by nature rule following particularly if there are penalties involved.
Suppressors are just another way for the gun industry to fleece gun buyers so I don't really care one way or another. As you said they don't really protect hearing (only over the ear protection can do that) so what do I care??
Assault weapons bans are, as you said, problematic for any number of reasons. Although the 10 year AWB had a marked affect on mass shootings (seems mass shooters often buy just before committing a massacre) no matter how they are defined the gun culture will change it to evade the intent of said ban.
Where I suspect we differ is in what is banned. Ban the feature, not the name. The features that make 'assault weapons' so lethal is rate of fire and ease of reloading. Therefore I would ban all semi auto (shoot as fast as you can pull the trigger) with feeding devices (reload in seconds). If we can't prevent shootings, at least make them less lethal.
BTW, where are you on your research into safe storage legislation? Have you found any you will support?