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It was coded in Java but also uses opengl.
No, since Minecraft was coded by Java, you can not use Adobe Flash Player. Sorry... Sources: I program Java, I work with Minecraft mods.
In the current release, while you can access Sponge in creative mode, it is not coded to do anything, or rather the earlier code it used to have, has been removed.
Yes, you will need Java. I recommend installing the latest version of Java, JRE7, which I have provided a download link to it on Oracle's site just on the development of JRE7, it will highly increase Minecraft's performance. Also, Minecraft is coded in the Java language. It must be read with a Java program, and in your task manager, Minecraft is "javaw.exe" so a big yes!
Zombies are hard-coded to always be hostile toward the player, so without using cheats or mods, there is no way you can tame a zombie, as such. However, you could consider trapping a zombie instead.