Yes, you can. Be careful though.
Yes. When you select multiple Labels while filtering cards, these are found using "OR"
(Open a board then Menu | Filter Cards to perform these search actions)
Example: Selecting Green and Yellow (you can name labels too) will find All cards that have Either Yellow OR Green attached.
Yes. When you enter words in the search text box. These are found using OR
Example: Search "Mobile Desktop" will find cards that have Either word Mobile OR Desktop.
Mixed, If you select any multiple options (labels, text in the Search box, select users or "Due in" options) these are combined using mixed AND and OR
Example: Select Green, Yellow and search: Mobile Desktop
Will return all cards that have Mobile OR Desktop AND marked Green OR Yellow
No. If you want to search "Doing" (or any) list only. I know of no filter to search a single list.
Once you get the hang of it, you can do some powerful searches.