In microsoft word, for paragraph formatting, you have this option:
"Don't add space between paragraphs of the same style"
Is there a similar feature in indesign?
The short answer: no.
The useful answer: you can do this by making use of "Next Style" in the Paragraph Style definition:
When you paste text, select all and in the Paragraph Styles panel right-click the "Lead Para" style and choose "Apply Lead Para, then Next Style".
For repetitive formatting of text in individual frames, a killer shortcut is to set one up as above, then make it a new Object Style. In the Object Style definition check "Paragraph Styles" and "Apply Next Style". You can then drop text into a new frame and apply all the formatting with one click on the Object Style.
Use your Styles to define your spacing before and after specific paragraphs...
You'll most likely need to create a custom style (eg. "No space") and define the paragraph spacing within that style. Then when you select that saved style for each of the paragraphs you've stacked, it will adjust the spacing between paragraphs as defined in the style.
One way to set your paragraph spacing...
You've defined that your Body style has no extra space. The only time there's extra space around Body is when it's next to something which isn't Body.
So put the space in the other styles — your Headline, Subhead, Scene, whatever. Any running Body copy will have no spaces. Any Headlines will have them.