Interesting range of questions here, which you may need to address separately.
Is booting from a freshly downloaded Linux CD safe?
Relatively safe, as long as you download from reputable source and check the hash you can be confident it hasn't been tampered with in transit. There is a small chance the source may have been tampered with, so you should be aware that you can't guarantee 100% security.
Can malware jump hosts via BIOS?
Theoretically yes - but I don't know of any in the wild that does this...yet. You'll be pretty safe here
How can I ensure the BIOS is clean
Not easily - a good BIOS rootkit could subvert updates and overwrites, so this will be very difficult to do.
Dual boot systems
What do you mean by this? Dual boot can be effective at separating the operating systems and is possibly preferable to running a CD boot each time you want to use Linux. Depends on the risk profile.
Back to your stated use - you want to do this to store Bitcoin on your computer. If you use your Windows OS to mine/trade bitcoins the added security of booting into Linux to store these coins is minimal - your Windows OS caches so much data anyway. So I wouldn't bother.