Hypothetically, if I use a computer with a linux OS, and let's say there is malware in the bios and the firmware and the hdd. And I download an .iso and transfer it to my usb-card reader using etcher, and etcher 'validates' my .iso transfer, and then unmounts immediately afterwards, as it does, would that mean the new linux OS on my sd-card is clean, or could it still be infected?

etcher just validates that the data on the card is identical to the data on the ISO file. It doesn't check for malware.

I don't know how your hypothetical infections can be transferred to the ISO file, but my ignorance won't protect you.

yes, it actually depends (may depend) on the malware itself. If that was able to do something tricky (unexpected) ...

If worried about an infected download, I'd create an md5sum of your downloaded .iso, then check it online, (that is, don't download the md5sum), against the genuine md5sum issued by the distro website.

How is it that you've concluded that you have malware in BIOS, in firmware, as well as the HDD?

1. You can upgrade BIOS by booting your system live and programming newer, or the same version to the BIOS. You just have to prepare all the files and utilities for this in advance.
2. What exact firmware are you complaining about? For instance on a motherboard, the firmware is the BIOS. Granted there are other cards which can have firmware, but you've not been specific enough here.
3. If you suspect a hard drive, you can set it aside and use an entirely clean one. You can also attempt to wipe a hard drive completely, but if you do not trust any wiping actions, then do not use the hard drive.

If you suspect that you have an infected system, and at several levels in that system, then deal with each component independently, clean them to your satisfaction, and the proceed to install a fresh OS. Further, do not use this system until you feel you have sufficiently cleaned it. If this is your only system and you have nothing else to use, then recommend you approach a friend, relative, or person you know who has a PC and see if they can help you download the utilities you need onto fresh media.

But what if your system is infected in such a way that md5sum (and other checking programs) are infected too and deliver the wrong (compared to NON-infected) answers.
With an infected system all bets are off.
Note that the OP was talking about a BIOS infection, so just booting up a live-system isn't enough, the only way is to check the iso on a NON-infected system (which on ITS turn should be checked first from an original live-system DVD, preferable NOT a burned one).