altiumdesign-rulesdrc
I've routed my PCB but had to change the minimum keepout rules which means I'll now have a lot of DRC errors – most tracks are too closely placed together and that would violate the new keepout rule.
However, I can see the errors only when I route a track. 
Is there a way to see all DRC violations at a glance? I'm sure there's a better way of doing it than selecting each track individually.
It's quite simply that Saturn has no idea what chips/passives are connected to the target line. If you look at the 2nd example, your driving voltage is 3.3 volts and it couples 1.65 volts to a track that is effectively equi-distant between source and ground. It cannot produce an answer other than 50% of 3V3.
Any driving node on the target line is going to look like a few tens of ohms and this will decimate any cross talk and convert what would have been a cross-talked voltage pulse to a current sunk into the driver.
As "The Photon" suggested, the situation can be solved by creating a special rule targeting specific only vias and pads or vias and vias of the same net. For example something like the rule shown below
Best Answer
If you have your rules set up properly, all you have to do is go to Tools --> Design Rule Check and click "Run Design Rule Check" at the bottom left of the pane. I tend to use the shortcut T-D-Enter and it does all of the above in just three keystrokes. This will list all of the design errors, provided you told the DRC to check them in the batch check. If you want to change what rules it looks for, go to Tools --> Design Rule Check (or shortcut T-D) and in the left-pane, under "Rules To Check", you can select your category and which rules you want the DRC to check.