Learn English – Look down upon vs look down on

From Oxford Dictionaries ( https://en.oxforddictionaries.com/definition/down ):

4.1 On or on to a list, schedule, or record. ‘I'll put you down for the evening shift’

So, "down to do something" means on the list to do something, or scheduled to do something, or otherwise committed to do something.

As for "down with", it's a well known colloquialism. From the same source:

US informal predicative Supporting or going along with someone or something.

‘you got to be down with me’

‘she was totally down for a selfie’

‘‘You going to the movies?’ ‘Yo, I'm down.’’

And from Merriam-Webster ( https://www.merriam-webster.com/dictionary/down ):

5 a slang : cool - a down dude b slang : understanding or supportive of something or someone —usually used with with trying to prove that they were down with hip-hop culture — J. E. White

6 : being on record - you're down for two tickets

You’re right: “Look it up in the dictionary” would be preferred unless the meaning of the sentence is “Look up the foreign dictionary in the library” or something like that. “Look up the dictionary” gets the meaning across, but it’s sloppy English.