Q: Do the conditions of skins I put in the trade-up-contract affect the condition of the resulting skin? Are the conditions of the skins I put into the arms-deal-contract or trade-up-contract affect the condition of the resulting skin?

A: No, they do not.
I have personally invested in 10 Factory New skins, only to get a Battle-Scarred skin as a result.
The only thing guaranteed in a trade-up contract is the Item's Quality going up by one "tier", to a max of Covert (from Case Collections).

A: It is impossible, or there is a very very low chance of getting a BS skin from a FN trade up. Putting in factory new skins gives a relatively high chance of getting MW or FN with a low chance of getting FT.
The skins used in trade up contracts do affect the output's wear. Here is a link where 1000 skins of different wear were traded up and you can view the results: 
http://steamcommunity.com/sharedfiles/filedetails/?id=420402033
