Generally, you only use present continuous to describe a situation where one thing happens while another is in progress, for example:
I was washing the dishes when the parcel arrived.
washing the dishes is a continuous activity, and the arrival of the parcel is something that happened while the continuous activity is in progress.
can is used to express a capability- normally something that is always true, like being able to write. We use present simple to talk about things that are always true:
I can write
In your first sentence, if you are expressing the idea that you are capable of doing the work, regardless of when it will be done, that is something that's always true, so you use present simple:
I can do the work.
You might consider using present continuous if you want to talk about doing the work while something else happens. Say two people are going on a business trip. One might say to the other:
I can be doing the work while you drive.
This version focuses on the doing of the work. If you instead focus on the ability to do the work, then you would still use present simple, because it's always true:
I can do the work while you drive.
Looking at your second sentence, we normally use present perfect to talk about something has been completed before now, and has a lasting effect to present.
I have finished the work - I finished the work some time ago: the work is now finished
You can use "can have" if you are talking about the ability to have complete something before some specified time in the future:
I can have done the work by the time we arrive.
may is different, since it is used to specify probability rather than capability:
He may be stuck in a traffic jam - probability about situation now
He may be waiting in the lobby - probability about situation now
He may be going to Paris next week - probability about a future activity
He may have been delayed - probability about something that happened in the past, with an effect that lasts to present.