"Proceed to doing something" or "Proceed to do something"

That is an interesting question, and I think the answer is that it depends. In the example you provided, I think both phrases could be considered correct, though I would definitely prefer the second one, because it reads much more logically to me.

'Now proceed to write on the paper' sounds like a command being given to someone in the present, telling them to begin the process of writing on the paper. Meanwhile, the first phrase sounds like you're telling someone to perform an action that is already being performed.

The way that I can see this working is if the person is already performing an action, and you're asking them to move on to another action: they were jumping on the fence, and you ask them to proceed to singing on the fence.

If you change the structure of the phrase to include the word 'begin', it reads a little better as well (I say 'a little' for a reason, since this is pretty bulky and awkward):

Now proceed to begin writing on the paper.

Overall, I find the usage of the '-ing' to be pretty awkward on the whole, and I'd say when in doubt, don't insert it. But it definitely can be used in some circumstances.

In the first case, to is a preposition. Accordingly, the phrase proceed to means to make progress by moving to the next stage in a series of actions or events, or to move in a particular direction.

http://www.macmillandictionary.com/dictionary/british/proceed http://www.ldoceonline.com/dictionary/proceed

Therefore, in my opinion, a more suitable sentence would be:

Now, proceed with writing on the paper.

In the second case, to is the part of the infinitive to write, and describes the action you have in mind (take a look at proceed to do something), though, it is used sometimes to express surprise or annoyance.