[susan53]

I'm not sure what you're asking about. Is it :

a) can despite/in spite of be followed by this/that?
As I said before : In spite of/Despite are prepositions and so are followed by a noun phrase (which includes pronouns). So - in spoken English probably that, as by the time the word has been spoken, the idea it refers to has passed, while in written English probably this as they're still on the page.

or is your question :

b) Would the sentences be syntactically correct?
No - only the middle one. You have two main clauses here and so they must have a conjunction to join them into one sentence. So either :
She wasn't well, but despite that she went to work.
or
She wasn't well. Despite that, she went to work.
As we saw in the last example, But can also be used as a sentence adverbial, so :
She wasn't well. But despite that, she went to work.
is OK too

NB : The word fine is only used in assertive contexts - ie those which assert that something is true. These are assertive contexts :

(How are you?) I'm fine / I'm sure that will be fine / These examples seem fine to me

The following are non- assertive contexts, and fine has to be replaced with other words - OK, right, well, all right, etc - depending on the context:

a) the speaker knows or believes it isn't true :

(How are you?) I'm not very well / I doubt if that will be OK / These examples don't seem right to me /

b) The speaker doesn't know if it's true and asks about it :

Are you OK? / Do you think that will be all right? / Do these examples seem OK to you?