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Step One - Applying modifiers to the enchantment level
The first thing that Minecraft does is apply two modifiers to the base enchantment level. Each modifier is restricted to a certain range, with numbers close to the middle of the range more common than those near the ends.
The first modifier is based on the item's "enchantability," which depends on the material and the type of the item (see the table below). Minecraft picks a number between 0 and the enchantability, then adds that number plus one to the enchantment level. Bows have an enchantability of 1 for this purpose.
Shoot...SCORE!
Source
1. Login failure
The login is failing because you haven't bought Minecraft for those accounts. When you buy Minecraft it will be listed as one of your games in your account when you log in on Mojang.com, but until then you don't have a Minecraft login, you just have a login to a website that doesn't do anything except tell you that you haven't bought any games to use it with yet. You can still play in offline mode, since Mojang is not so obtuse as to require you be always-online in order to play Minecraft, unlike some other game publishers. However...
2. Server connection failure
When you connect to a server it will check if your account has a Minecraft registration too, and will drop the connection if not. So there are two times your login is checked for Minecraft access: once when you start the game, and once when you start multiplayer.
Fixing the server connection
The server's behaviour can be modified by editing server.properties and setting the line:
online-mode=true
to false instead. The players wanting to connect then pick a name to use consistently (since the server won't enforce identities to match what it has saved in its world files) and "try" to log into the game with that, let it fail, then play offline. Since the server is set to offline mode too, it will allow them to connect under that name.
The problem with this is that anyone can connect – make sure that the server isn't exposed to the internet, and the worst of the complications will be avoided.
The more likely problem though is that you'll have a bit of a headache migrating the kids' work once they have full accounts, since it's likely that the names they pick won't be available to choose once Minecraft is added to the Mojang account (which is when you can pick an official profile name for use in Minecraft). If there are name changes, then you'll have to either walk the child through the idea of losing their inventory and location on the server (blocks they placed will still be there), or you'll have to fiddle with NBT editors to transfer the inventory and location from the old name to the new one in the server's save files. Not hard, either one, but a stumbling block to be aware of.
The real solution here is to register a Mojang account for each child and purchase Minecraft for each account. I understand the desire to test it all out first, but if they already like Minecraft in single player, there aren't really any surprises that come with multiplayer that should change their or your mind. The advantage here is:
You'll immediately get the names that they'll be sticking with (consult and choose wisely, for children are fickle!) and can avoid identity issues in the server's save files.
You'll be able to set custom skins for their avatars, which children inordinately enjoy, in my experience. These are kinda nice in single player, but since they're readily visible in multiplayer they become more relevant when playing with others.
Fewer steps in logging in and connecting to multiplayer where mistakes could happen (a registered account can be remembered with it's password, making logins one-click), which for younger kids can be a bit of a headache-saver for you.
My kid has had an account for a year or so now and loves playing with other people, even if it's only to make random block stacks and randomly-enchanted gear in Creative mode. But then, that's 5-year-olds for you.
Best Answer
Yes, but with rapidly diminishing returns
According to the wiki page on Enchanting1, the effect of Unbreaking is:
You can see that the formula is different for armor, and even with the regular three levels, the chance to not lose durability when hit increases slower with higher levels. Extrapolating the "20%/27%/30%" using the formula given yields:
(Values are rounded in the table, but not in game). The chance converges to 40% for high values of Unbreaking, with diminishing returns the higher the enchantment level gets, to the point where is negligible.
If you directly compare an Unbreaking III Diamond Chestplate to an Unbreaking IV one, on average, the former breaks after 529*100/(60+40/4)=756 hits, the latter after 529*100/(60+40/5)=778. That is only a relative increase of 3% more hits.
1 While the page has changed, the game mechanics have been the the same since at least March 2013.