Modern druidry doesn't necessarily involve the worship or any particular set of gods, unless you are in an order like ADF, which is essentially a church. I assume they have certain gods, but I am not entirely sure. I am in the Ancient Order of Druids in America and the Druidical Order of the Golden Dawn. The AODA has members from all over the religious spectrum, including Christians. Druidry isn't necessarily a religion. It can be practiced as a spirituality. While some druids, such as myself, invoke gods and goddesses (usually of one Celtic pantheon or another), some don't use any gods, preferring to involve energies of the elements. My religion is Wicca, which is complimented by my druid spirituality.
As far as the circle, they can be similar, but aren't the same, and in fact, the way you work with the energies has a different feel. Every morning, I do AODA's sphere of protection ritual. While it has similarities to calling the quarters, it is a bit different. Instead of creating a circle in which to hold energy for a purpose, it is meant to call in the elements (actually seven energies in all) and surround yourself with them in order to keep out negative energies throughout the day. (You can read a sample ritual at https://aoda.org/Articles/The_Sphere_of_Protection.html
As part of my DOGD practice, I perform the Lesser Ritual of the Pentagram every evening, which invokes the elements using certain divine names and involves projecting a circle similar to casting a circle, but again for a different purpose. (You can read about it in The Celtic Golden Dawn by John Michael Greer.) You can work within it, especially meditation work, but whereas I cast a circle for particular purposes in witchcraft, whether it is divination, ritual observance, or to do spellwork, in my druidry practice, the circle IS the purpose, at least more often than not, and it is left in place all day long rather than drawn up at the end of the ritual. To further complicate things, we use the same rituals to set up a grove or temple similar to Wicca in order to celebrate the sabbats. I am still fairly new to druidry, but that has been my experience so far.
It's important to remember that modern druids do not have the same practices as the original druids, although we are inspired by them and a number of our rituals are at least partly based upon what we know of them, the reality is that there is very little surviving information. There isn't an actual, direct connection. That being said, modern druidry is three hundred years old, at this point, and covers a wide range of practices. I practice magic, but there are many druids who don't, at least not to the extent of what we would expect as witches. Each order approaches things differently. In AODA, we have druids who rarely if ever do magic outside of the sphere of protection. They are ecologists, permaculture farmers, artists, etc. ADF operates as a church with a certain set of beliefs. Yet, the Gnostic Celtic Church within the AODA does not require any particular pantheon. OBOD is more of a spiritual philosophy (from my understanding. I am not totally familiar, though I plan on joining them as well at some point). In the DOGD, we are all magicians and invoke Welsh gods according to the system of magic designed by John Michael Greer and based upon the teachings of the Hermetic Order of the Golden Dawn, but modified for druid practices.
Modern druids do a great many things, which makes the answer to the original post pretty complicated. On each point, I could say some do, but some don't. We've incorporated a lot of practices that aren't even Celtic in origin, including runes, tarot, gi gong, and eastern meditation practices. Personally, I tend to worship the Wiccan god and goddess concept and often Celtic gods such as Bridid, Lugh, Dagda, The Morrigan, and The Cailleach. I think it is common for druids to associate with Celtic gods, but it definitely isn't required. In AODA, we are encouraged to incorporate whatever gods (or lack thereof) that fit our religious beliefs. I invoke the above gods in my work. Mostly, we expand our spirituality through a close relationship with nature and a clear understanding of our impact upon our local environments. I find that it combines well with my witchcraft practice, since I tend to draw upon elemental energies.
I am afraid I just muddied stuff up more than anything else, but there you go. Modern druidry is sort of all over the place, but I think that is part of its beauty. It continues to evolve, much like Wicca. In fact, Gerald Gardner was a member of a druid order and was good friends with Ross Nichols, who founded OBOD. From my understanding, before Nichols, druids celebrated the equinoxes and solstices. Nichols added the cross quarter days, having discussed the concept with Gardner. As a result, in AODA, which predates Nichols, the cross quarters are optional, though I celebrate them anyway due to Wicca.
That seems like a lot of stuff, I know. You can find a lot of information on the websites for OBOD (http://www.druidry.org
) and ADOA (aoda.org). They were a lot of help to me when I was decided whether or not to add druidry to my practice. Otherwise, if you have any questions, I will be happy to answer them as well as I can. Like I said, I am pretty new, but have worked very hard to absorb as much as possible and will be happy to share my experiences thus far.