No. The last version you can get with this launcher is 1.5.2. You can't even play 1.6 snapshots with it.
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You can log into the Minecraft launcher on one PC with as many accounts as you want. You can also install as many as you want on each computer you have. However your saved games are not syced between PCs. On the same PC though, each launcher will still be able to access the same saved worlds.
Minecraft is accessible on Windows, Mac and Linux. If you want it for Mac go to this link: https://s3.amazonaws.com/MinecraftDownload/launcher/Minecraft.zip Note: you still require a valid minecraft account.
To fix it, restart minecraft, and in the launcher, click the options button next to the login button and click force update, than try again
You can still add mods using a different launcher through Demo/Single player you just can't join multiplayer servers. You have to purchase a premium account for that.
The Minecraft launcher needs to go online to download the game the first time you log in on any given computer. After that, you can play Minecraft on that computer without being connected to the Internet.Multiplayer won't work offline, though.If a new version of Minecraft is released, you must go online to download the update. You can still play the old version offline before you do this.