Hello thar, and welcome

First up, I'll answer the MP question. It doesn't cost a dime, zip, nip, nada, nothing to play online. The servers are all run by Freelancer fans/players.

When you try to access the Multiplayer - make sure you permission the Freelancer.exe to have access to the internet, otherwise you will never get to the server screen. Once then, choose servers via good ping and number of players. If you see a padlock next to it, then its a locked server and you need a password (may not be a public server, so may not be able to find passwords).
Ones in grey with red version numbers are modded servers, where you may require a mod to play on.

When you first connect you can sort the server - preferably by either ping or player count. Whatever you do, I would always stick to servers with a ping lower than 100 if possible. Anymore and you may experience bits of lag every now and then.
You get your MP account number when you first connect, but this is stored ni your Freelancer directory as well, and you only need to note it if you plan on uninstalling your game and then reinstalling later... wishing to keep your same characters. Characters are saved on servers, and your MP account ID identifies whom you are, and which characters are yours .

As for Alaska, some servers may run server side mods (you don't require anything) that unlocks the Alaska jumpgate - but there isn't anything in there too do apart from fly around without any company. There are no npcs, no bases you can land at - and nothing to do/shoot/loot.

Some mods (client and/or server side) may add populations to Alaska, but I would recommend playing standard Freelancer before having a go at the mods!