I cannot figure out how to sync bookmarks and settings onto a new device from a previously synced profile
I had a PC that had all of my bookmarks and such synched to my Firefox profile. Unfortunately that computer's HDD crashed, I'm trying to find out if I am able to "download" the old synched data to a new PC, but I can't seem to find how to do so. Is this even possible or are my old bookmarks lost?
Thank you for your question about the Firefox Sync service.
It appears from your question, that you are using Firefox Sync as a back up service. This is not what Sync is designed to do. The Firefox Sync service takes a copy of the data you wish to include and transfers it to a second device (typically a mobile device such as a tablet or a telephone) running a copy of Firefox. The storage in between all attached devices is both temporary and fragile and is not stable enough to be (and is not designed as) a reliable backup service.
If you have a copy of your Firefox profile for desktop Firefox, you may be able to recover your bookmarks and other data. Please have a read of this article.
If you have a copy of your data on a mobile version of Firefox, please follow these instructions to connect it to Firefox Sync to copy your data.
Beyond what Seburo said, you might be able to log in to your Firefox Account on the "new device" to recover your data thru Sync - but if you get a message about the password not being correct do not reset / change the Firefox Account password! That will wipe your data from the Sync server due to the data sitting on the Sync server being encrypted via the 'original' password thus making it useless when using a different password.
That said, if you have the old HDD, even if Windows won't boot, you might be (or should be able) to recover your old Firefox data via a SATA or PATA to USB adapter like this: https://www.amazon.com/AGPtek-Drive-Adapter-Converter-External/dp/B00BIE996S Worth $13.99 to experiment with, IMO.
Used a few different like that over the last 20 years myself to transfer data from hard drives that wouldn't "boot" by the operating system that was present; the hard drive did work as an "external device" to recover important data from the hard drive piecemeal.