Yes, Multiple Choice and Dropown fields both support multiple answers. You can turn this on by toggling "Can choose more than one answer" in the question's configuration.
Can I let submitters choose multiple answers for a multiple-choice or dropdown question?
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Can I let the user type in their own answer for a multiple-choice or dropdown question?
Yes, for both question types. Toggle "Include an 'Other' option to specify another answer" in the question's configuration.
How do I add a dropdown field to my form?
On a blank line, type "/dropdown" and select "Dropdown" from the popup menu, or click "Add questions" and select "Dropdown" from the question dropdown.