Do you think ancient deities have changed over the centuries or is their perspective of things still rooted in ancient times?
Every thing changes. If we do not change and adapt, we die. The gods do not exist in a vacuum. They are a part of the world, and from what I can tell they live and learn in a similar fashion to us humans.
How do they relate to issues of the modern world?
How do humans? We all take in the issues as we are exposed to them and react according to our own natures. Deity who are involved in the healing arts are aware of the advances and failures of modern medicine. Those involved in warfare know about the technology of their field as it currently exists. Agricultural deity know about the advances in farming methods.
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)?
I've never encountered a deity who wasn't able to relate to the here and now. I've encountered two who weren't as up-to-date on the state of the world as is generally the case, but both were the sort of deity whose primary drives haven't really changed since the long ago and far away, and they, much like their human counterparts, don't have much need to modernize their point of view. While the mechanics of farming has changed, the motivations and goals are pretty much the same as they were when the first farmer poked seed in the soil and prayed for a good harvest to feed their family. My local land's spirit responds to the same things now that she did when my ancestor's lived in the area.