Is it normal for tap water to turn brown/blue when the pipes freeze?
We're seeing here in action why backflow preventers (check valves) are mandated in various municipalities.

Blue water sounds like someone was using one of those toilet cleaner tablets in the tank, the system lost pressure due to the freeze blockage and siphoned the toilet tank till the fill tube on the ballcock was exposed. Not something I'd want circulating in my drinking water. It also could be copper sulfate from corrosion that occurs in copper piping, but that usually is only enough to stain sinks and tubs.

If the water is orangish brown, it may be that the stress from freezing has knocked a lot of iron rust loose in the pipes. Typically, there is always a major air gap between sewage and potable water (nigh impossible to get the toilet bowl to empty into the flush tank due to gravity and four inches of air the water would have to jump).

Your major worries with pipes freezing is that if they do burst, they can draw contaminated water back in if they're in places like flooded basements and the act of draining causes a vacuum within. At that point, you have bacteria contaminated water to deal with.

Your landlord definitely should be notified and should tell you what the water contaminant is.