Can you collect coins in the spartan objective?

You can, and even red and blue coins, just not power-ups.

Yes, all the coins are fine, but the spartan objective is to not use any power ups, angel wings, or anything besides what was already mentioned. It usually takes about a little over four thousand meters depending on how many coins you collect. Everyone says that when you get a million you need to kill yourself in case you accidently use a power up after since it won't work. It's sometimes impossible to avoid the power ups, but there usually is a way either move to the side or don't jump or slide. It took me about twenty tries, but just keep trying because it's like trial and error to see what works.