1. What is the best way to set trace widths for each net?
Open the Rules & Constraints editor (Design.. Rules..) then select the Routing and Width rule. You get to set the minimum, preferred and maximum widths. When placing the track, press 3 to cycle through them. The cycle also includes the last custom value, so if you hit TAB and use the properties page to change it to whatever you want, that value will be part of the cycle.
In the Rules editor you can add a new rule for particular types of track. For instance, a 'power' net class. Then placing nets of that class will use this rule instead of the default.
but it doesn't fill or connect as shown in the screenshot.
The screenshot isn't telling us anything (except it doesn't work). Right click and there should be a 'Violations...' entry in the context menu. Select that and it will be more informative (albeit possibly obscure). Tell us what it says if it doesn't help you.
In 3D view the board is translucent or hollow looking
Open the View Configuration panel, select the View Options tab and check the transparency options group.