It doesn't really have to do anything with egy. It's very very rare that consonants get turned into other consonants while building words. (gy -> g)
Egész + ség + ed + re
Complete + [noun] (completeness = health) + your + onto = "(on)to your health!"
This is also a prime example of a word which contains one each of all three suffix types. -ség is a képző (changes the word's core meaning), -ed is a jel (gives more information about the properties of the word), and -re is a rag (defines the grammatical role of the word).
I guess after all the flying kindergarten teachers we've been yearning for some normalty. :)
Zebra can also refer to a pedestrian crossing, because of the similarly black-white striping. So it may be a more common occurance in Asia to see a tiger jumping onto a crosswalk because he's all about safety in traffic.
"The tiger jumps the zebra"? Okay, it might make sense in English, especially if you use a word like "pounce", but ugrik is, like most other -ik verbs, (almost) strictly intransitive. It doesn't take direct objects.
Also you'd need a definite form of the verb in your example.
I would not say that, sorry. Many "-ik" words can take direct objects, with either the definite or the indefinite conjugation. Just a few examples, very basic ones:
"Eszik" - "almát eszik" - "eszi az almát"
"Iszik" - "almalét iszik" ...
"Játszik" - "játékot játszik" ...
"Vacsorázik" - "zsíroskenyeret vacsorázik"
Maybe these are indeed the minority, and most "-ik" words are indeed reflexive/intransitive. (Update: yes, that seems to be true, the "-ik" type of conjugation used to belong to reflexive and passive verbs. Later it got extended to other verbs. (Source))
But I can even add a direct object to "ugrik":
"Nagyot ugrik", "hármat ugrik", "fejest ugrik", "ugrik egy hátraszaltót", "most ugorja a tripla szaltót", etc.
No, the problem is something else. It just does not make any sense to say "zebrát ugrik". How can a jump be a zebra?
But you can jump over a zebra, which can be expressed with the accusative:
"Átugrik egy zebrát" or "átugorja a zebrát".
But to jump onto something, that needs "-ra"/"-re".
And to jump a car, you need a "bika" in Hungarian:
"Bikázza az autót" - "jumps the car".