Note: The following is from the Chicago Manual of Style, and is for American writers. The Brits (and others) have different opinions about quotation marks.
There is not a hard and fast rule, as far as I can tell, about putting dialogue in paragraphs. It depends on the context. If the dialogue is a logical continuation of what came before, then it can be in the paragraph. If someone else speaks after that, it should be on the next line and indented. If the dialogue is more of a departure from what came before, then you should indent it.
Amy walked into the party. It took her three seconds to realize that her Victorian gown was going to stand out among the other guests, who were dressed in jeans or motorcycle leathers. “I should learn how to read an invitation correctly” she said to no one in particular.
“I like mutton sleeves,” said a tattooed woman in red leather who was standing behind her with a drink in her hand.
Always use double quotes in dialogue, unless you have a quote within a quote, in which case the quote within a quote has single quotation marks around it.
“Why did you call me ‘a motherfucking asshole’?”
Long paragraphs of dialogue, such as when Lord Winterberry finally declares his love for the Lady Ethelgrange after 300 pages of slowly building romantic tension, has a quote at the beginning of each paragraph of the long declaration, but no closing quotes until the end of the declaration.
“Lady Ethelgrange, it behooves me to be short. I love you more than the stars, more than the waterfall, more than the fleas on the royal dogs. I will love you until the sun falls from the sky, which will never happen, because it will probably go supernova first.
“Your eyes are blue jewels in your face. Your skin is soft as velvet. Your fingernails are perfect ovals of dead tissue. I love every hair on your head and under your arms. All the other ladies—and a few gentleman, I might add—mean nothing to me any more.
“You are the wind beneath my wings, the cobblestones beneath my carriage, the horseshoes beneath my hooves, the song in my heart, and the garlic smell on my breath. Will you marry me?”
(OK, stop gagging, I’m done now.)
According to the Chicago Manual of Style (13.41) unspoken words, such as thoughts, imagined dialogue, etc., can either be put in quotes or not as per the author preference (or house style of your publisher).
What do you say?