It is safe to run the stock intake without the cover on (it will just get dirtier a bit faster).

You can use a hose to clean off a stock filter but I wouldn't recommend it as the stock filters are essentially paper which can be broken down by water. If you do use water, make sure the filter is dry before putting it back on. You can get a K&N replacement for your stock filter if you don't yet want to go with a full intake then you won't have to worry about cleaning the stock filter every so often.