Do you think ancient deities have changed over the centuries or is their perspective of things still rooted in ancient times?
How do they relate to issues of the modern world?
I agree with Melamphoros. I don't see why deities can not change and grow as the world and universe do. Nothing in Nature is stagnat and they are apart of Nature, so this tells me they change with the universe.
Even though this world has changed since the ancient times, many of our worries are the same, just in different context. Many people have to work about money and jobs... of that falls under prosperity. There are still many many deadly illnesses and diseases... so deities of health and healing are probably still busy.

Do they have archaic/ancient characteristics or perspectives you find sometimes hard to deal with (or do they find it sometimes hard to deal with your modern perspectives)?
Hmmm... well the only thing I can think of is how some war/battle deities and so darker deites were said to like some bloody and unpleasant sacrifices. Now whether or not this is true, I don't know, most of the information on the Celtic gods that wanted those kinds of sacrifices were give to us by Ceasar and he isn't always a reliable source.
I personally would not sacrifice any living creature to any deities.