If your hatchery is full, I know you automatically get another hatchery tab, but let's just say that you had 26 adolescents in your hatchery, and you move one to your stables, will you lose a hatchery page? I know it doesn't really matter, but I'm curious :)

Hatchery pages automatically appear and disappear in relation to how many pets are in the Hatchery, so if you have 26 pets in the hatchery (2 pages- one full page and one slot used on the second) and one pet is in some way removed from the hatchery- whether by becoming adult, being sold, whatever- then it removes the superfluous page. If you then bred yourself a new egg, then it would add the second hatchery page again to house it.
Same if you have 51 pets in there-3 pages (two full, one with one slot used) where losing one pet will drop your page count to 2 pages, or 76- 4 pages (three used, one with one slot used), removing one pet will drop you to three pages.

Aka, yes you 'lose' the page, but it vanishes because it's no longer in use.

Thanks!