Absolutely, the chain must be as wide as your thickest cog. Many single speed drive trains are 1/8" in which case you need a 1/8" chain.

Some single speeds have narrower chain rings and rear cogs, in which case a narrower chain will fit — although you could use a 1/8" chain if you preferred.

Too much flexibility is not a problem. A derailleur system needs side to side flexibility. As a single speed pulls in a straight line all the time, the flexibility is irrelevant: neither harm nor benefit.

Many derailleur chains are designed to facilitate shifting. This will neither help nor hinder on a single speed.

Personally, I'd say buy a cheap 1/8" chain because it will fit any single speed set up.