It looks like your pan needs a hard cleaning and a re-seasoning. I run into this about once every 2-3 months (depending on how much I use it).
My tried and true method is to break some rules at this quarterly cleaning. I'll scrub with an abrasive scotch-brite, SOS Pads or similar scrubbing pad. During this time I often use about a quarter cub of baking soda, and towards the end I'll even use a little bit of dawn dish soap to make sure I've removed all of the particles, which amount to fine iron filings.
I've even resorted to very high grit sand paper to even out some areas in very serious issues on my handle. Now it's important to rinse and dry the pan COMPLETELY.
I then carefully rub crisco (less smokey than lard) all over EVERY surface of the now shiny and smooth pan.
Then, invert pan in oven over a baking sheet or foil to catch any mess, and bake the pan at 500F for roughly an hour or two, or until it is uniformly black.
If you have access to a clean, wood burning put, that's what my grandmother swore by, but I find the 500F oven more consistent.
I am not certain if this would work for your high carbon steel pan pictured in the first photo though. I would read up on that.
It might be better for you to understand some of the factors involved, and make your own decision how to treat your pan. Below is some basic information I think you will find helpful.
Background: I've successfully reclaimed old, rusty pans, fixed a few that family put in the dishwasher (TERRIBLE IDEA) and so on, and maintain my own regularly.
Cast iron is "seasoned" when it develops a highly carbonized and oxidized surface. This surface coating is both harder and more slippery than the plain iron.
This surface only truly develops with 3 things: High heat, carbon and oxygen in the presence of your pan.
The heat can be provided by a stove, oven, campfire, grill, torch, etc.
The Carbon can be provided most effectively by some kind of oil. I prefer crisco/shortening most. Baconfat second, but that's just an opinion.
The third thing is oxygen. That's in the air, so it is both free and plentiful.
Now, other things to consider. If you have an unseasoned, rusty or poorly seasoned pan, it actually might be best to "start over". Using some kind of abrasive (steel wool, sos pad, even light sand paper) you can remove any unwanted coating, until you get to the grey, dull iron underneath.
Then apply oil, and apply high heat in the presence of oxygen.
Sometimes this coating may take several applications to develop, and will actually get smoother and darker with use. Rubbing with things like salt and whatnot can actually act to kind of polish the surface.
And the best part is, if you truly don't like how an application turned out, scrub it raw, and start over. You'll be removing almost immeasurable amounts of the iron. I have my GREAT GREAT Grandmother's skillet, and it has been seasoned and re-seasoned perhaps a hundred times in its life.
The rest is mostly guesswork, opinion and conjecture.
I hope this helps.
Best Answer
I would never leave it unseasoned.
Carbon steel rusts. If you just cook with sufficient oil, and let it cool down with the oil still inside, and then clean gently (preferably without any soap), the inner bottom will indeed session itself with time, as tfd mentioned. But the rest of the pan will be exposed to the air and will create a layer of rust. If you don't use it frequently enough, or if you prepare acidic food in it before it has gathered some seasoning (and most vegetables are acidic), you will get rust on the bottom too, before seasoning has had time to build up on its own.
Ingesting rust is not an immediate health risk, but it is better to avoid it. Our bodies weren't designed for it, and you might end up with too much iron in your diet. It is also porous, so the pan would be hard to clean well. And it looks really shabby to have a rusty pan in the kitchen.
And if this is your first "normal" pan after years of non stick, you might be amazed at the amount of oil needed to prevent food from sticking. It looks terrifying to our modern "fat is bad" eyes. And you still have to be careful, despite the oil. With time, you will learn to cook the right way, but having it seasoned first certainly helps.
There is no law requiring you to season, but to me, the above reasons are enough to always season.