How to know if I can rely on an app to keep my password?
You have to trust the developer of the app. Period. 

Besides code analysis there is no way to eliminate the possibility that an app does send your password somewhere. 

And even tools like Little Snitch won't help you if the app is allowed to send emails from your account. 
If you send an email to somebody it would just use the established connection to afterwards send another email with your password. Since the connection is encrypted you can't see what the app is doing. And because the whole purpose of the app is sending emails the connection is not suspicious at all. Same thing is true for all apps that need access to the internet, if you enter the password to an facebook app it could just use facebook to send a private message with your password and so on.  

To make it even more difficult to detect, the developer could implement it in a way that the malicious routine waits 30 days before it sends your password. And if the developer is a smart criminal (he wrote a great app so he probably is) he can detect testers who just fast forward their device date setting too.
If the developer is smart it is nearly impossible to figure such things out without looking at the code. 

Entering your password somewhere is all about trust.
And with all those apps from unknown developers it's hard to figure out which app you can trust enough to give it your password.
That's the reason Twitter, among others, uses an authentication process where the app never knows your password. 

Having a different password for different services helps in case you have accidentally trusted an untrustworthy developer.
But your email password is a little bit special because usually it can be used to gain access to almost all services you use.