Mysql – Backup remote database using thesqldump

I am trying to achieve a quick way to do a backup and download a remote mysql database to my local machine, using just the terminal.
The remote server uses ssh keys for authentication.
I can do it by running seven sequential commands on the terminal, some executed in the server, others from my local machine:

1) connect to remote server using ssh key;
2) dump the database to a remote directory;
3) logout from server;
4) download the backup to my local machine;
5) connect to server again;
6) remove the backup from the remote directory;
7) logout

#1 ssh -i my_rsa_key my_user@my_domain.tld
#2 mysqldump -u my_db_user -pmy_db_password my_db > my_path_to_backup_directory/backup.sql
#3 logout
#4 rsync -chavzP -e 'ssh -i my_rsa_key -C -c blowfish' my_user@my_domain.tld:/my_path_to_backup_directory/backup.sql  /my_path_to_local_directory
#5 ssh -i my_rsa_key my_user@my_domain.tld
#6 rm my_path_to_backup_directory/backup.sql
#7 logout

Question 1: is this workflow ok or is there a "cleaner" way of doing this?
Question 2: can I automate these commands in a single batch, for me to just type one command in the terminal and get all seven steps done? I have tried chaining the commands using

&&

but I guess it doesn't work since I am executing part of the commands from my local machine (such as rsync) and part from the remote (such as mysqldump).

All help appreciated,
thank you.