Can I do transactions while Bitcoin Core is being synchronized? How long does this process take? It is telling me 46 weeks left but its doing this for months. How do I speed of the synchronization?



I downloaded it from their website. It is the latest version of Bitcoin Core. Windows 10, 4.00GB 64 Bit
I assume, you are using bitcoin QT client, then the general answer to move coins without bitcoin core being sync'd is yes :-) Simply use another wallet software. You could export your keys from the current bitcoin core system, and import it into another wallet software. I am doing this often, and I have written my own unix shell scripts to then create a transaction manually. However I recognize, that you are a new member here, and I assume, you do not have the experience yet to do so. So it is quite difficult to explain the process, without having a sound understanding, what you do. 

Why it is so slow in sync'ing the blockchain? There can be many reasons: 
I explained it here: https://bitcointalk.org/index.php?topic=4500995.msg40584294#msg40584294
There is no easy way to speed up things, but if you have a decent QUAD CORE CPU and 4 gig memory, with an SSD, you're good to go a few days.

For the sake of quickly getting things done, playing with low volumes (low values of funds), I guess the best option for you would be to export the PRIVATE keys from bitcoin wallet (there is plenty of messages in the forum her, just search for it, keywords QT wallet and dumpprivkeys), and then import into some SPV wallet, or some online wallet. However, there are different representations of keys to import and export (keys can be hexadecimal, or WIF or even WIF-compressed). So there is a learning curve involved! You would need to find a matching pair. Perhaps the question should be rephrased a bit, and someone in the community can help out on compatible export/import wallets...