Automatically update stock prices
GNUCash can be told to update prices on startup, but if you're doing it right you don't open it daily. Just for reference / sanity's sake I have a cron job that updates stock / fund prices daily.
Cope with stock splits
There is a stock-split tool for this.
Cope with stocks in more than one currency
If you just mean owning international stocks, this isn't an issue. GNUCash allows one currency type per account, but child accounts can have different currencies than their parent. So each fund / stock you own gets it's own account, bought and sold in whatever currency you prefer.