-Does Electrum require my password to send funds to another address? (I don't want my family to be able to access my Bitcoins.)
Yes, you set a password as part of the wallet creation process and that is required to spend bitcoins. However, there is no restriction on viewing transactions or the wallet balance.
-Can I do cold storage?
Yes, but you may want to get familiar with hot storage, if I were to coin a phrase, first. When you are comfortable with that you can try cold storage:
https://electrum.org/tutorials.html#offline-mpkCold storage is not for noobs. Not just noobs to bitcoin. But also noobs to electrum. At the very least you need to know the difference between wallet seed, password and master public key (the answers are in the docs on that site).
-Is Electrum really a safe client to use? (Up to like 5 BTC)
Yep.
-Is my wallet secure?
It's as secure as the rest of your computer. Don't download and install crap. Keep the OS up to date, install an anti-virus and keep that up to date , make sure you have a firewall etc.