Is it possible to play Freelancer as a pirate from the beginning of the game?
(I’m already playing without story missions)
I tried landing on Rochester, but there weren’t any missions available.
If not, why would anyone ever buy the Bloodhound or the Dagger?
Speaking strictly about single player vanilla Freelancer you can’t. You have to play the story at least as long until you have access to the jumpgates. If that is too much story for then that means: NO.
How to get around those restrictions
You can however play multiplayer (even if it’s just a server for you) or install appropriate mods to lift those initial restrictions.
As for your apparent problem with faction standing: Every faction has friends and enemies. By shooting members of one faction you not only anger them and their friends but also please their enemies. Determine which enemies your favorite faction has and shoot them (For pirates try shooting the police). It takes a bit of grinding and a good enough combat ship but it is feasible.
If you are interested in mp may I suggest you check out the Mod “Shattered Worlds – War Torn”.