All the while her thought was trying to justify her delight in the colours by merging them in her mystic religious joy. (1.1.36)
Dorothea is so caught up in her spiritual life that she feels guilty about her "delight" in earthly things, like horseback riding or the "colours" of her mother's emeralds. This "delight," though innocent, is something that she feels the need to "justify." So she tries to persuade herself that the "delight" she feels in looking at the emeralds is "mystic" – it will somehow help her soul get closer to God.
The intensity of her religious disposition, the coercion it exercised over her life, was but one aspect of a nature altogether ardent, theoretic, and intellectually consequent. (1.3.13)
Dorothea's "religious disposition" has more to do with her spiritual life than with the doctrines of a particular religious sect. But her spirituality isn't always described as purely positive, it "exercise[s]" a "coercion […] over her life." In other words, her spiritual feelings coerce, or force her, to make decisions of which her intellect or common sense might not approve.
Will was divided between the inclination to fall at the Saint's feet and kiss her robe, and the temptation to knock Naumann down while he was adjusting her arm. All this was impudence and desecration, and he repented that he had brought her. (2.22.32)
Many characters associate Dorothea with saints, angels, or the Virgin Mary, and Will is no exception. But Will doesn't associate her purely with the realm of the sacred. He's "divided" between his impression of her as something sacred (hence the desire "to fall at the Saint's feet"), and his sense of her as a physically beautiful woman who can inspire jealousy (hence his inclination to "knock Naumann down while he was adjusting her arm").
"I should like to make life beautiful – I mean everybody's life. And then all this immense expanse of art, that seems somehow to lie outside life and make it no better for the world, pains one. It spoils my enjoyment of anything when I am made to think that most people are shut out from it." (2.22.50)
Dorothea explains to Will Ladislaw why she can't appreciate art the way he does, and her reason stems from her spiritual beliefs: she wants to "make life beautiful" for everyone. This is why she doesn't care for art – "most people are shut out from it," and it therefore doesn't do anything to make the world a better place.
"I have a belief of my own, and it comforts me. […] That by desiring what is perfectly good, even when we don't quite know what it is and cannot do what we would, we are part of the divine power against evil – widening the skirts of light and making the struggle with darkness narrower." (4.38.29-31)
Dorothea tells Will about the core of her spirituality: intentions matter more than outcomes, and good intentions are part of a universal fight against evil.
There is no general doctrine which is not capable of eating out our morality if unchecked by the deep-seated habit of direct fellow-feeling with individual fellow-men. (6.61.33)
Spirituality is privileged over religious doctrine in the world of Middlemarch for this reason: it's possible to follow "general doctrine," or the rules laid out by a particular religious sect, to the T, and yet still be a bad person. Doctrine only helps if it goes along with sympathy, or "direct fellow-feeling with individual fellow-men." According to this novel, if you follow all the rules, but don't care about other people, what's the point?
"What do we live for, if not to make life less difficult to each other?" (8.72.4)
Dorothea sums up her personal doctrine (as opposed to institutionalized doctrine laid out by organized religion) in this passage. The whole point of life is to make things easier for each other. And the fact that Dorothea uses the first person plural ("we") suggests that she expects everyone to act as nobly and generously as she does.
"This young creature has a heart large enough for the Virgin Mary." (8.76.49)
Here's another place where Dorothea is compared with something sacred. Mr. Farebrother compares her generosity to that of the "Virgin Mary," but this isn't how Dorothea sees herself.