score:2
Welcome to Christianity.SE ! Since you're a veteran of the SE networks you will soon realize that you will need to do some work to make the question acceptable to this forum.
It looks like you are new to Christianity and are sincere in trying to understand it more. So as a Christian who has been around (i.e. know the framework, not saying I'm better morally than you) I try to provide a general preliminary answer which for now I hope can help guide you to edit your question.
First, "grace" is a highly technical theological term that different denominations define differently (just see wikipedia), so you'll have to tighten your question accordingly (see the tour). But for the sake of this answer, I'll just define it as the undeserved gift from God to us so we can overcome the effect of the Fall to attain the goodness of life that God intended us to have.
Secondly, theologians will distinguish different types of grace for different kind of gifts. Understandably different denomination will categorize them differently. To specifically tailor this answer to your questions, I'll divide it into 2 kinds:
Armed with the above working definitions, here are the answers to your questions:
I heard that it is through the gift of God's grace that we have the impulse to do good and that it is always with us - but the source that I heard it from was questionable.
Most denominations will say that the impulse to do good, let's call it "goodwill", is an inherent part of human nature, which in turn is created by God to do good. Therefore, it's part of the original design, and thus can arguably fall under "Common Grace" for denominations who define grace that way. If that's the case, the impulse (although marred by sin) will always be with us because God will preserve creation until the Day of Judgment.
But denominations differ on whether the acts resulting from that goodwill can be properly called "good" without the existence of "Special grace" and without the person being "born again". For example, in the Catholic schema, a good act is "meritorious" only under certain condition.
Thirdly, most denominations will say that "Special grace" is needed for salvation, so no one can save him/herself with "Common grace" alone. Also, most denominations will say that "free will" (an aspect of "Common grace") will always be with us to make us responsible to accept / deny Christ despite the free will itself has been heavily compromised due to sin.
So this is a question of applicability.
- In Christian theology is it possible to be without God's grace?
Since life itself is due to Common Grace, it is not possible. For Special Grace, I think all denominations will also say that we will not survive in our sanctification journey without grace. In addition, in Catholicism you can lose your "state of grace" due to mortal sin; so if you die in that state you don't go to heaven.
- Are there places where it is withheld?
In denominations that believe in predestination, Special Grace is given only to the elect.
- People for whom it has been withdrawn?
Depends on the denomination. But most will blame it on us, not on God's initiative (for example, see here). Basically, if one decides to continue further and further into sin, this is what St. Paul say God will do: Romans 1:18-32.
- Do animals have the same grace?
Special Grace is only for human beings. Animals enjoy God's goodness of creation, but we do not even categorize it under Common Grace. In general the term "grace" doesn't apply to animals. God may have a "redemption plan" for animals, but it's not revealed to us, so we don't know. Humans are supposed to care for them, though (per mandate in Genesis 1:26-28), so we need to treat them well while using them for our benefit. So if they DO receive "grace", it's through their "official caretaker": humans.
- Do the various graces always apply albeit in different ways and intensities?
In Special Grace, different denominations will define the dynamics differently, mostly in response of the Christian's engagement with God and the Christian's spiritual life. In Catholicism for example, there is a well-defined subdivisions of "special grace", sacraments to confer them, spiritual states of a person affected by different types of sin, etc. This is where denominations will give wildly different answers.
I hope the above helps. I will edit the answer once the question is more specific.