"Dash" is used also in recipes. For example, "add a dash of salt". It means a quick, brief toss - just a little bit and you're done with it. A "dash for the door" is the same idea. It means a quick brief movement towards the door, and you're done with it. You can "dash" somebody's ideas. It refers to when somebody demolishes your arguments with some quick brief comments. Ugh! That hurt. Somebody dashed my beautiful idea.

DASHED FOR THE DOOR is fine. Maybe not the most usual way of saying it, but it has a certain style. MAKE A DASH FOR is kind of a phrasal verb I think. In general you can use DASH just like run, as long as it's a short run. Short races are even called DASHes. e.g. He finished first in the 100 meter dash.

‘Run for’ is fine. When the bombardment started, I ran for cover. (To find cover) ‘For’ is based on the idea that there is a reason. Otherwise ‘to’ is more natural when talking about a destination. ‘Dash’ is often used to mean ‘go away quickly’, without a destination. For example: ‘Dine and dash’ means order food at a restaurant and leave without paying for it. ‘I have to dash’ (I have to go quickly) But ‘dash for …’ is ok, again with the idea that there is a purpose for needing to get somewhere quickly.