The expiry of BN weekly options is every Thursday.
Yes you can hold monthly BN contracts. The expiry for this is the last Thursday of the month. There are no restrictions on the contracts you can hold. You can hold weekly contracts along with the monthly contracts.
The charges for trading BN options is the same, whether it is monthly or weekly contracts. There use to be an STT trap for exercised ITM BN options but that has been rectified now. Read more here.