If I include an image to a question (e.g. using my Dropbox account) will they be cached? So, can I remove them from my Dropboc public folder after posting the question without removing the image in the question?
Nope. There is no caching being done (in a reasonable amount of time) by the system. Any and all images not hosted by StackExchange's (paid) imgur account are susceptible to breaking.
It should be noted, however, that upon inserting links through the button interface of a q/a, the image is re-hosted immediately.
I suppose a fair way to do this would be to 'ask the system' as it were.
Consider the example:
which shows me this as I type this answer:
I will remove the Dropbox-hosted picture in five minutes (to give the system time to cache if it does so).