ok, one by one:
Silkscreen is the layer which contains whatever markings you want to have on your board (like, component designators, component orientation hints, high voltage warnings, hints for troubleshooting (eg.g 'this should be at 1.23345V' ), wild african animal pictures, design revision number, website, your name, basically whatever you like to)
Soldermask is the thermoresistive lacquer which covers all of the pcb excluding soldering pads and usually vias. This is to prevent oxidation/degradation of copper and to solder components more easily/reliably. What is actually specified in Gerber files are openings in soldermask rather than mask shape (so you mark where soldermask should NOT be).
Solderpaste works similar way as soldermask. It depicts opening that you would have to make in screen printing stencil to print proper amount and pattern of soldering paste. Stencils are usually done only for volume production.
I don't use PADS so I dunno where to look for these options. Generally in most software Soldermask openings are defined along with pad definitions and design rules. You can of course add additional soldermask openings (for whatever reason, e.g. sometimes silkscreen is not used so you can place an easily readable part number on pcb as opening in soldermask)
Solder paste stencil openings work in similar way. In simple cases (like standard 0805 passives) they are more or less similar to soldermask openings. In more complidcated cases (especially heatsinked packages like TO-252 and TO-267) there may be need to "chop" stencil opening into smaller pieces to provide thinner layer of solderpaste when component is placed.