To my knowledge, it's customary for artists to charge based on how much work is involved. So, they'll calculate a wage for themselves and multiply it by the hours of work + material cost. Usually, that wage is relatively low, i.e. close to minimum wage, if your country has that.

So, you can try to roughly estimate based on that. You'll probably forget some of the work needed to actually create it, but at the same time, artists are usually also quite quick at their craft.

However, you can also ask for a rough estimate upfront. It's not going to be that artist's first rodeo, they'll probably have a pretty good guess, and they understand that people do care about the price.
And if something unforeseen happens, which means significantly more work is required to make your idea happen, then they'd hopefully also inform you about that and offer you cheaper alternatives.

Having said all that, if you are getting art made, then it is a good idea to have money to spare, even just for your personal finances.

If you are working with a specific company to produce a personalized / custom product, they will generally have at least one person on staff that can do design, or at least deal with any file conversion and technical stuff like that. They should be able to quote you a price.

If you are working with an artist first, they may have a company they're familiar with to have the item produced, and if they do, should be able to handle most of it themselves. If they don't, or you would like it produced somewhere specific, just let the artist know - most places will have all the information the artist will need on their website, or the artist can just get in touch with them directly.

In general, if you're going to an artist or design shop they should be able to give you some kind of quote with a very rough idea of what you want - especially if they do this professionally, talking to people who don't have an art background is basically half their job. If you want to be specific it's much more useful to send something like a mood board or a collage of similar things than a bunch of text. It's also generally a good idea to be open to their ideas, as they often have experience that may be helpful and will see some problems much quicker than you might.

On comission sites there can be an average price for each artist; you can then add the complexities you want around the art and increase the price based on how difficult it can be in your head. Most of them also list how much art corrections go for, or how many are included.

As for the file format; most hat manufacturers will accept PNG format; but you can also have the PSD in case they don't so you can convert it easily.

The internet is vast, and many somebodies make hats. You will probably find it’s much more expensive then a normal hat, as the prototyping cost is spread against many hats when you buy one.

You can select an artist online, and work out an hourly rate and an estimated time. The more work you have done on the hat, the less time you will spend on them filling in gaps in getting to a finished product. If you are forming felt, that will require building a mold, which can take a bit of time and money.

I mean, at the end of the day, I just want a hat, and at least a guestimate on its cost.

I'm in awe of the direct links to anything. However, man, people assume the top result of DDG an impressive amount.