Wow, a blast to my childhood here! I remember reading the series you've described as a kid back during my dragons phase. I even still own the first book.
This is The Pit Dragon Trilogy written by Jane Yolen. The books are Dragon's Blood (1982), Heart’s Blood (1984), and A Sending of Dragons (1987). A fourth book, Dragon’s Heart was published in 2009.
On this world, days are like ours but nights get very cold, requiring anyone stuck outside to find a shelter until dawn (not the false dawn)
This is the Dark After on the planet Austar IV, where the books take place. During a four hour period in the night the two moons overlap and it becomes so cold that humans cannot remain outside without dying.
The other thing I remember is that in one of the books by slicing open a dragon and spending the night, the character emerged with special dragon powers.
This is detailed in the book Heart's Blood:
Golden, Jakkin, and Akki flee to the mountains, but Golden is badly
injured. Pursued by the Wardens, they are forced to leave Golden
behind in a cave which they had sheltered in during Dark After, using
Heart's Blood's body as a door and heater. Jakkin manumits the
traitorous Erikkin, and Heart's Blood dies from a stinger shot in the
neck, her body saving Jakkin and Akki. The only shelter they have
during Dark After, is Heart's Blood's body, so they crawl into her
birth sac. In the morning, Jakkin and Akki are reborn out of the
dragon blood, and became the first real human Austarians, linked to
each other and Heart's Blood's five hatchlings.
Jakkin and Akki are now able to communicate the way the dragons can, due to their "rebirth" from Heart's Blood carcass.
And yes dragon's blood is acidic. Quoted from Dragon's Blood (link withheld b/c not sure if legal):
Blood Brother gave one last mighty pull and his wing tore free, the
hot blood dripping down onto the dust, burning the floorboards where
it fell. Three drops spattered onto Jakkin’s back, leaving deep pits.
But he had been unconscious since the first blow and did not feel the
burns.
...
The orange dragon in the pit shook its head and the blood beaded its
ears like a crown. A few drops spattered over the walls and into the
stands. Each place a drop touched burned with that glow peculiar to
the acidic dragon’s blood. The onlookers ducked. One watcher in the
third row of the stands was not quick enough and was scared on the
cheek. He reached up a hand to the wound but did not move from his
place
