File size when using FULL recovery mode

Hi everyone, 

I have set up a AlwaysOn cluster with one database. The database 20 GB. I have two disks, the MDF file is on disk 1 (50 GB) and the LDF file is on disk 2 (20 GB).

Before the AlwaysOn Availability Group can be created, the database must be set to FULL recovery. 

I created a maintenance plan for a backup at night to a network share.

When I came in this morning the 20 GB disk was full, the replication stopped and the database was unavailable.

I deleted the logging database to get things to work again.

A couple of questions:

1) Is it true that LDF file will become as big as the MDF file when using FULL recovery?

2) Can I shrink the database so the 20 GB disk will not be filled up?

3) Do I need to extend the 20 GB disk?

Thanks!