Some of the things you've mentioned here you actually can test for even on closed-source implementations. It's pretty easy to trace the activity of the app to see if it creates temporary files or does network activity so you can you investigate that stuff.
As for the other things, like using the rng properly and whatnot, no, you can't really check that stuff. But your implication here is that open-source apps can be trusted because you can verify that stuff, and I don't buy that. Unless you yourself are a crypto expert that's qualified to carry out such an audit, and you have the spare time / inclination to perform a full audit of the app, then there's no reason why it being open-source should make it any more trustworthy. Perhaps if some independent trustworthy third party performed the audit you could then decide to trust it, but closed-source apps can still be audited, it just requires the help of the app developer to do so.
And of course even if the app is audited (whether open- or closed-source), that audit will only really verify the particular version that was audited. Future changes may introduce vulnerabilities again, so unless someone qualified to do so is constantly auditing all future changes, then you can't really trust it anymore, since your trust model is that the source needs to be independently verified to be trusted.
On other hand, if your trust model is that you determine whether you trust the people involved to get it right, then it doesn't matter if the app is open- or closed-source, as long as it's developed by the right people. Granted, it can be hard to determine whether someone can be trusted to get it right without independent audits, but speaking personally, I take tptacek's "I feel like they know what they're doing" recommendation as carrying a fair amount of weight. I certainly would welcome an independent audit of 1Password, but I recognize that I can't really expect a closed-source software vendor to hand the source of their flagship application to a 3rd party.
(if it isn't clear, I'm a happy user of 1Password)