How can I transfer a Bitcoin-qt wallet to Electrum? I am switching from bitcoin-qt to Electrum, and it seems as if I will need to download the blockchain again. Can I copy and paste my wallet.dat into Electrum's files?
According to this FAQ:

In Electrum 2.0, you cannot import private keys in a wallet that has a seed. You should sweep them instead.
If you want to import private keys and not sweep them you need to create a special wallet that does not have a seed. For this, create a new wallet, select “restore”, and instead of typing your seed, type a list of private keys, or a list of addresses if you want to create a watching-only wallet.

You can get the private keys from Bitcoin Core using dumpprivkey as described in Using dumpprivkey to make a paper backup.