I've rephrased the question's issues some; I hope that's okay.
1. Do the spells animal growth et al. come into effect if the subject's new space is littler than the space available?
Yes, because that's what the spell says happens. Although normally "[i]f you ever try to cast a spell in conditions where the characteristics of the spell (range, area, or the like) cannot be made to conform, the casting fails and the spell is wasted" (Player's Handbook 171 and here), spells in this vein can come into effect even "[i]f insufficient room is available for the desired growth, the creature [still] attain[ing] the maximum possible size" (198 et al.).
2. What does "constrained without harm" mean when the embiggened subject fails to break its enclosure?
So far as I'm aware there's no official guidance on what this means beyond what it says. Were I the DM I'd rule that the subject creature is squeezing, the subject suffering appropriate penalties depending on its new space and the dimensions of the enclosure as per on Squeezing (PH 148) or, more expansively, on Cramped Spaces (Drow of the Underdark 159). I don't know what other rules would be used beyond house rules, especially since "the spell[s] cannot be used to crush a creature by increasing its size" (PH 198 et al.). According to such spells, the subject always reaches it new full size, and it's always unharmed when expanded into an itty-bitty enclosure. That's magic for you, I guess?
For example, the world's most abused lion could be pushed to, essentially, take 20 on an Escape Artist skill check and—with four folks successfully aiding it—take 20 min. to squeeze into a 5-ft.×5-ft. adamantine box (DC 30). After a helper shut the door, some jerk wizard could cast the spell open/close to open the 1-ft.-diameter porthole in the side of the box so the 5th-level Sor/Wiz spell animal growth [trans] (PH 198) can be cast on the kitty. ("An otherwise solid barrier with a hole of at least 1 square foot through it does not block a spell’s line of effect" (PH 176).) If the lion fails its saving throw then fails its Strength ability check to break its enclosure, the lion would be—somehow and remarkably—unharmed inside that box despite its size category now being Huge. (I suspect it will make the full extent of its discomfort clear upon its release.)
3. What if, at the subject's new size, part of the subject's new space would be occupied by one or more other creatures?
Again, the game offers little in the way of guidance except insofar as it says that typically during combat a creature's space can't take up even part of another creature's space. (Y'know, unless they're all really little or one's a lot bigger than any of the others—there're always exceptions.) However, on Accidentally Ending Movement in an Illegal Space, in part, says
Sometimes a character ends its movement while moving through a space where it’s not allowed to stop. For example, you might incur an attack of opportunity from a monk while moving through a friend’s square and become stunned. When that happens, put your miniature in the last legal position you occupied, or the closest legal position, if there’s a legal position that’s closer. (PH 149)
While I know that changing size so as to occupy another creature's space isn't moving, like, at all, no other rules I could find applied better, so extrapolating from these seems to me a reasonable choice. I mean, I wouldn't have the subject creature make a Strength check to break the other creature or whatever—that's odd and would require a whole new batch of mechanics.
Instead, I'd just rule that the that the embiggened creature squeezes, suffering appropriate penalties (see above). I'd also rule that it must continue squeezing until an area big enough to accommodate it is available—an area that doesn't see its bigger space overlapping with the space of the other creature or creatures that were present when the size change magic came into effect. I'd even rule that the subject creature could stay squeezed where it is and make its attacks… while suffering the appropriate penalties, of course. (Still, though, even this simplified process becomes increasingly convoluted with further additions to the scenario.)
Sharing Spaces: A Bad Fit
I wouldn't do it, but a more adventurous DM might adapt the Rules Compendium's rules on Sharing Spaces (95). Doing so, though, is so complicated that I can't really know how a DM would go about adapting those rules. Those rules are geared toward what happens when, during combat, a helpless, prone creatures becomes not helpless (helpful?) while one of its squares is occupied by another creature. Those rules sort of seem to urge the formerly helpless creature to start grappling with its space's other occupants, yet the rules don't seem to consider, for example, what happens if two or more occupants are in the bigger no-longer-helpless creature's space. There's some material for a DM to riff on, but how to—and whether to!—riff on it will be up to the individual.
I mention the Sharing Spaces rules then not because they're good but because they exist. A DM that's looking for inspiration for his house rules because—for some reason—this is happening all the time should be aware of all the resources that are available.