First of all: setting up an autorouter to produce quality results requires so much experience and efforts that in almost all cases the results compare rather poorly to a manually routed board. It can be useful to quickly lay out non critical tracks on the board, but most experienced PCB designers will tell you to take stuff in your own hands. Autorouters tend to ignore harnesses and buses, and thus create random patterns with excessive vias (thus increasing trace resistance).
That said, Altium allows you to define almost anything PCB related as a PCB design rule. Some rules are automatically generated with every new PCB, but not all of them are critical. In your case for example:
- The minimum solder mask sliver defines the minimal width of solder mask strips between the pads of ICs. Smaller solder mask strips simply can't be fabricated, which means that there won't be any solder mask between the affected points. The default is 10 mil however, which means that most IPC compliant SMD footprints like QFN and TSSOP will always violate this rule. If you apply an appropriate amount of solder paste, then ICs can also be soldered without a solder mask between pads, so don't worry too much about this. This rule is particularly useful to set between high voltage tracks because having no solder mask might allow these tracks to arc over.
- Silk to Silk: no idea why this rule was even invented, it tells you when overlay objects like component outlines and designators are too close to each other. It has a purely aesthetic purpose, and won't affect the functionality of your PCB in any way.
- Silk to Solder: allows you to set a minimal distance between overlay objects and solder mask edges or exposed copper. It's useful to set this rule to a small value for the entire board, like 6 mil for example, to keep overlay artwork and text far away from exposed pads. This allows for some variations in the screen printing process, while protecting your copper pads from getting covered in paint and thus cause trouble during soldering.
If your board has two layers (with tracks on the red layer = component side and on the blue layer = solder side) then you definitely have a two layer (double sided) board. By "layers" PCB designers mostly refer to the electrical layers of a board, not the overlays, solder masks, solder paste, keep out, anything mechanical, ... So a single sided board is a board with one copper layer (either top or bottom), a double sided board has a copper layer on both top and bottom, a 4 layer board has a top and bottom layer plus 2 internal layers, etc.