Is it a typo that badges will go on sale in February, but the hotels will be available in January?  How will people be able to get a hotel without it being linked to a badge?

I don't think the hotels will be available for booking through the house portal in January. The way I read it is that they will only give us details about the hotels and portal in January and the actual booking process will come later after badges are available for purchase. 

Gen Con badge registration begins on Sunday, February 9.

Housing block (hotel) reservations begin on Sunday, February 23.

"HOUSING REGISTRATION
Details for Gen Con hotel registration will be announced in January 2025."

Here's an example from last year that I got in January (Jan 12 specifically, but it'll be different every year. Also, it's a website, but it's exactly the same thing we got through email). This announcement itself is about badges, but note the section at the very bottom of the page:

https://mailchi.mp/gencon/01102023-1934910?e=31c82e5f1a

Screenshot of the bottom of the page:

I don't know if this is what it's going to look like this year, or if there will be a different one specifically for housing itself, but the point is that as long as you're signed up for the newsletter, you'll get emails that'll contain the important upcoming dates, and explanations of what to do.

So yes, in January, you will receive details about hotel registration. The actual registration will occur at a later date.