first a wireless door bell made by arlec find in any h/ware shop second maybe find a old power/gas box and convert it into a mailbox third lockwood sells a keypad lock for gates and does,nt run on power at all you find it at your local h/ware store also

you can buy battery operated door chimes.
they’re wireless.
a lot depends on how far from the gate is to your house is.
they run from about $30.00 on up.
think i got mine at home depot or Lowe’s. for sure one or the other.
they have a whole section for door bells.
you may need to watch out with what you want to do for your mail delivery. ask your postal deliverer if they know of any restrictions for what you want to do. the postal service has rules as to what you can and cannot do.
good luck to ya.

The doorbell is a simple 2 wire unit, should be 18 ga wire up to a total of 100′ (50 each way) further go to 16 ga wire to allow for line loss.
A keypad entry system is probably best installed by a locksmith who is used to remote entry systems. Talk with them before you install any wire for the system. They can probably do it easier and faster than you.
As to the mail box, try the big box store in your area, and talk to the post office.

Check at Home Depot in the doorbell and mailbox section and ask someone at the store with help. If not you might want to buy a book from Home Depot or Lowes about this project.