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Fellow, having third stroke, shows sign of life when leaving hospital (5)

F r ank(h)
Ankh - symbol of life

More concise better?

Fellow, having third stroke, shows sign of life leaving hospital (5)

Out in the open at Coney Island, perhaps, going no further without husband (5)

Coney Island is a frankfurter
Frank(furt h er)

I don't think I've ever heard of it as a "Coney Island," merely a coney. The term does derive from the hot dogs, wieners, franks, whatever you like to call them, sold at Coney Island. I may be wrong about all this. Perhaps some of my countrymen have further information?

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Follow nose to foul-smelling bard's pigsty (5)

F(ollow) RANK. Def in Chambers.

He did it his way, honest! (5)

mprs, I lived in Brooklyn for about 7 years and, though I can't give citations, I'm pretty sure I've heard and seen the term "Coney Island" hot dog / sauce / etc. Though "Coney" is also common.

There's a Wiki reference, some online recipes, etc.
For a menu example, https://www.leosconeyisland.com/Menu/Coney-Specials

In New England we always call them Coney Island hot dogs.