About your first question, you can automate the start of the application by creating a file: /etc/rc.local and put there the command that you want to run on startup:
About your last question, you have two options to do this.
Option1: A reverse proxy like nginx. This aproach is better if you have several services in this machine. There are several tutorials on the web to do this.
Option2: Use the embebbed tomcat in the application for your certificate. This is the easer way. You can follow this steps:
Use certbot to create your letsencrypt certificate. (You need python installed):
git clone https://github.com/certbot/certbot
cd certbot
Generate your certificate:
./certbot-auto certonly -a standalone \
-d <your_domain>
Generate PKCS12 files from PEM files:
Go to /etc/letsencrypt/live/<your_domain>