I am a system level developer who has implemented encryption technologies used in top-secret environments. Also I have worked on mobile device development at a system level for many years. I can't detail my credentials, but for as much as anyone else on Slashdot can be considered reliable, ... well you take it from there.
1) So far as I know, the only "smart phone" OS which has been "properly audited" was the specific versions of BlackBerry OS which is used by Obama. This does not include all versions of Blackberry OS... only the versions which have been specifically audited and approved for use on his phone. This does not mean that the OS is secure, the NSA audit on the code was performed too quickly for my tastes. It just means that the majority of "obvious holes" are not present. This completely rules out the newer QNX based OS for Blackberry since there is absolutely no possible way that much code could be properly audited in the time which it has been available. On top of that code audits are only a small part of what you need to do to secure a few million lines of code which is heavily communication oriented. Of course, running a simply security auditor on the OS helps as well, but I wouldn't bank on that either. An OS needs years of testing at a single revision before it can be truly solid.
2) Android may or may not be secure. It's extremely unlikely. If however you want Android and can't live without it, make sure to use only OS images which are hash check verified (MD5, SHA...) from Google directly. If the phone can't run the stock OS, DON'T USE IT! The reason for this is that the OEMs often update and modify code before putting it on the phones. They are feature oriented, not security oriented. Google Nexus would be a decent choice for this.
3) Don't even consider Windows, Symbian or iOS based phones. iOS is the safest of those three, but lacks pretty much all the features you're interested in. So far as I know, Apple doesn't even care about a "trusted platform" as the cost of maintaining a trusted platform is WAY TOO HIGH and would never yield the profits Apple demands from products. Windows and Symbian just aren't about trusted in the first place and the serious short comings in the Symbian "Development process" make it far too susceptible to being able to be hacked. Without decent development tools and kernel level debugging (which Symbian simply lacks for the most part) it's not possible to harden an OS. Also since Symbian never made use of things like "Test driven development", any change in one place could wreck 100 things elsewhere ... and often did.
4) Never EVER allow users to install apps... ESPECIALLY ON A JAVA PHONE meaning Blackberry or Android. This is because Java is insanely easy to hack. Yes, I know Oracle and Sun say otherwise... but I recall Yugo also calling their cars safe. Voluntarily installing an app which replaces the class loader on the system is enough to hack the entire thing. There are hundreds of other ways to hack Java which is obvious to me an others that can be exploited with a simple malicious chunk of code in an app. Also, since Java based platforms don't generally allow sandboxing, the apps all kind of have access to override system resources and interfere with each other.
While I personally despise Blackberry having tried it a few times and felt like I was using junk, if you must have these features, you should use their hardened and audited system.