Can someone else sign into my iTunes account to download my purchased apps?
Yes you can. And Apple actually doesn't discourage this behavior. While it's true they do warn users not to share their Apple ID, but that can be taken more as a security concern than one of theft. In the past, Apple has never been one to mince words about piracy (being vocal against music and movie theft).

Apple allows you to register a small number (5) of devices to a single Apple ID. So you can register your little cousins device, your girlfriend's device, or your dad's device without a care. 

This helps consolidate purchases and in a lot of cases, helps device management in large households. 

The drawbacks aren't many. You can withhold your password from those that you do not wish to make additional purchases using your ID. But this will restrict them from even installing free apps. And if they happen to log out, they will be unable to log in again. The decision how to handle your account's access is left to your discretion. Just note that you will be responsible for any purchases made on any device registered under your account. Typically Apple is quite good with refunds, but I'm not sure how they would handle one in this case. (It's likely that they will issue you a "one-time" refund and warn you that you will be responsible for any additional purchases made in the future.)

Aside from potentially having someone rack up your account, it's totally cool with Apple and a great way to get mileage on your app purchases.