I think its grammatically incorrect to combine "de" and "um" into "dum". When people speak quickly it probably sounds like "dum" though! I don't know why you can shorten some combinations like "de" + "isso" = disso but not "dum". Ah well, every language has annoying things like this I guess. :)
Some verbs always demand a particular preposition (which often remains untranslated because the equivalent English verb doesn't need it). Two good examples are:
- "gostar" which always comes with "de" - "Eu gosto de você" (I like you) or "Ele gosta de jogar" (He likes to play);
- "precisar" comes with "de" when followed by a noun - "Ele precisa de uma caneta" (He needs a pen) - but doesn't when it is followed by a verb "Eu preciso ir" (I need to go).
In American English we do say "Now, get under the covers" to children, but in context, we do use "cover" for "blanket" or even for a flannel sheet. "Cover" can absolutely mean other things, but it is not wrong when we are using it to mean blanket as many people do. Some people hardly ever use the word blanket.
It is the second definition for the noun here: http://www.merriam-webster.com/dictionary/cover Of course they put the verb first, so you have to scroll down a lot to get to the noun.