Is this a function of the radar physically "seeing" things, or that the pilot chooses the settings to show what they are interested in, suppressing returns that are just clutter and noise for the job at hand?
Radar, and radar tuning, is a surprisingly complex game. Some (military) radars allowed a great deal of adjustment, whilst most are quite limited. Computerised systems remove most of the complexity, but at times have a nasty habit of either removing the interesting returns, or of showing way too much. Modern airliner radars are automatic, and very few people go back to the base settings and play with them. You simply don't need to...they do their job well enough in auto. I like to fiddle with them because my background involved a complex system, so fiddling is ingrained. The A380 generates a 3d database of radar information, so, to a degree, you can look behind. Cool, and sometimes useful, but it's not really a presentation of the present, but rather a view of what it saw a few minutes ago.
What weather features can you see? Clouds, I guess, but are they differentiated at all? More subtle things like jetstreams and turbulence?
Radar has to reflect from something. That can be an aircraft, land, or moisture. Not air. Jetstreams and turbulence are basically air. There are turbulence displays (and warnings) within the system, but they need to be reflecting off moisture to able to gain doppler and density data.
Weather is generally colour coded by the insensity of the return, and we learn to interpret that. Simplistically, green is ok, yellow so-so, red bad, magenta horrible. But there are times when red can be ok, and green not....
As for other aircraft, there must be some way of knowing where they are located. Do you trust in ATC, look out the window, keep track on the radio ("Breaker, breaker, this is Roo Two, what ya haulin', good buddy?"), or do you just plow on through God's wide night sky out over the Pacific, relying on the hand of the Almighty to keep everyone apart?
Why? Keeping us apart is ATCs reason for being. If you don't trust ATC to do so, then you're going to be a very nervous wreck. There are times in non controlled airspace where we look after it ourselves, but mostly the sky is very full, and having an aircraft do it's own thing as it avoids imaginary issues is not a safe behaviour. TCAS provides a nice last ditch defence. As for radio calls..there are very defined rules as to just what is said, and when, and there is no place for 'breaker'.