I had to do a lot of speculating about Odo's internal anatomy in his solid form before I could even begin to write a story about him carrying and giving birth to a baby Changeling.
The way I see it, his 'lungs' are just air sacks and his 'breathing' isn't for oxygen so much as for temperature regulation and to generate speech, and out of habit he might catch his breath because he sees solids do it and he does it too to fit in. His surface is solid like ours, but I think underneath he's still very thick fluid, like Jello. When he 'breathes' he basically lowers the density in his upper torso to suck the air in and he raises it again to exhale. In theory he could puff himself up like a puffer fish if he wanted to! (But he won't, as that's just silly!) His process of "breathing" is such a habit he doesn't even think about it, just like we don't think about taking our own breaths every day unless something happens to hamper it, like asthma attacks, chest infections and so forth. It's also said that Odo's morphogenic matrix is its most malleable at a certain temperature, so the thermoregulation he gets out of "breathing" helps him stay somewhat close to that temperature so he can shapeshift quickly if he needs to. I imagine he might have to work harder to shapeshift on a boiling hot or freezing cold day. That's why he changed the temperature in his quarters when he was in labor--he knew on an instinctive level that he needed the temperature just right to make it a little easier.
When he was pregnant the baby was in a little bubble not much different than the bubbles in bubble wrap. It needed that environment to practice a few shapes until it was large enough and able to hold a shape long enough to survive its own birth. When it turned Odo back into a shape shifter, it created that tiny space in his chest for itself. It was microscopic at first and unaffected by him regenerating. Then it reached a size where it would be exposed to the outside too soon if Odo liquefied, hence him having more and more trouble regenerating until the night he couldn't turn into a puddle. That was the baby holding some control of its environment. (Poor Odo, though! He WAS fine with it once he knew the reason!) I chose the spot where his heart used to be for symbolic reasons, since all moms keep their babies in their heart.
Odo's birthing process was strung out on purpose for drama. Like I said before if he just relaxed it would have been one squeeze that's no more uncomfortable than a sneeze. Pretty boring eh? Since Odo is Odo and Odo likes things a certain way, he can't live without his routines and he needs order, and his body was creating chaos. The "contractions" were Odo's internal density increasing around his chest cavity. The increase started in his back and moved forward and downward, which was squeezed off his 'lungs.' Odo woke up feeling the pressure just beginning. He mistook what was happening as just one more "weird sensation" related to his condition. Basically, the baby was trying to make his body shape shift enough to let it out, and Odo was able to resist it all day. I based this it off how he resisted going all puddle on everybody when the Founders made him sick in Broken Link. (And there was a point where it didn't work and he melted anyway.)
BUT...to put it into perspective...if you're not used to doing hard physical work and decide to spend the day making a cinderblock fence, it'll be so hard to pick up that ten pound bag of cat food later at the store, won't it? Your arms will be sore and your muscles won't be able to handle the stress after spending so much energy on something much harder. Odo's body was doing the equivalent of lifting weights under the surface because he kept resisting the process. The pains he had were not from the waves themselves, but from his exhaustion after resisting them all day. His body was able to squeeze itself a little tighter and stay that way longer each time before he could force it back to normal. He could have kept going that way all night too, but it would have been a longer, more painful and more terrifying process to him and the baby Changeling. As he relaxed, the "waves" became less painful. They weren't waves like we think of them, but more like waves of his internal structure getting a little thicker and staying that way. Remember, he was exhausted from fighting it all day, so his body had create density, rest, create more density, rest...etc. The space around Kejal was contracting smaller and smaller. I did base that part off what happens inside a uterus as it's working on bringing out a baby, except in Odo's case his body would stay that way and just put more pressure on after a moment's rest, whereas a human uterus has to relax to let blood flow back into the placenta. The increasing density is what the tricorder would be showing if I chose to describe that, but I couldn't because the story is written in third person limited POV, and we're "sitting" behind Odo's shoulder.
The whole point of the story was for Odo to experience letting go, both physically and emotionally. He had to give up control. That's why I had Kira there. She is someone Odo trusts with all his soul. Kira got Odo to relax enough to get out of his own way. His tension and fear were going right to the baby, and that made it not want to be born because Odo's constant resistance made it think the outside world is terrifying and painful, so during part of the labor process(when Odo was writhing around in pain and yelling "damn!"), the baby began to resist too. Once Odo let go of the fear and let his body do its thing, he also let Kejal feel safe enough to emerge. The air pocket didn't break until Odo stopped fighting himself. The action of his internal density increasing forced the baby into that "toothpaste" shape to help it emerge. And when Odo actually delivered Kejal, they worked together--Kejal slithering out while Odo's body gave one big push in a manner not much different than wrapping your fist around a toothpaste tube and squeezing the toothpaste out. His chest went mostly liquid, but his back was almost hard as a rock.
As for the pseudo-endorphins, they're a side effect of all the 'tired' shapeshifting he went through. His finally relaxing meant his morphogenic matrix didn't have to work as hard, so the enzymes his morphogenic matrix produces when he(in my speculation) shapeshifts or links with other Changelings started to flood his system so fast they bonded into a new molecule--the pseudo-endorphins--and that is when he went from feeling no pain to feeling really good
all over. The pseudo-endorphins built up so much that when Odo surrendered control and delivered Kejal, he quite literally had an orgasm*, or the Changeling equivalent to one. In fact, Kejal felt it too, as they were linked until Kejal's body separated from Odo's.
I don't think of it as sexual at all, not as WE know it anyway. The ecstasy he experienced came from the processes within his body, from being linked with the baby as he brought it into the world and his feelings for Kira all coming together at once. The emotions and sensations he felt while giving birth let him bond to Kejal like any mother bonds to their child when they're allowed to hold and cuddle it right after birth. Odo was still feeling "high" when he picked Kejal up, said hello and smiled. And Kejal knew how much Odo loved Kira--and that is why at the end of the story he chose to have Kira's eye and hair color. Odo and Kira were the first faces Kejal saw clearly, so he mixed them together from memory when he took humanoid form.
But for Odo to experience birth like that, he had to be completely relaxed, open and accepting. He couldn't do any of those without feeling safe, and Kira helped him feel that. Without her he would have been tense and focused more on Doctor Mora and Doctor Bashir and worrying about everything except what he was supposed to be doing. Kira saved him from going through a terrifying and traumatic birth. She could understand how vulnerable he felt. She was his guide and his armor, and in my little headcanon it's part of what makes her want to know him as he really is by the time the episide "Chimera" aired.
*It IS possible for human women to experience this, but it's rare! http://pregnancy.about.com/od/births...asmicbirth.htm