How can I automate Time Machine?

Automating Time Machine is not a good idea. Its archived backups wipe themselves out each subsequent backup it can't find space to maintain them. You would actually need indefinite space on your backup drive to avoid that happening. It helps having at least twice the capacity on your backup as your original to avoid too much initial data loss if data is frequently changed on the original. Furthermore, an automated Time Machine backup does not give you the opportunity to backup to a second destination. It is better to restore your Time Machine backup to a second drive occasionally, in case the machine loses its hard drive, because you never know Time Machine backup's own consisntancy until a restore is attempted. And if its consistancy fails, you are left with nothing.

A third reason not to automate your backup, is it can frequently happen at inopportune times when you need more machine horsepower. And this slows you down. A manual Time Machine backup avoids that problem.

Yes it is possible. Apple menu -> System Preferences -> Time Machine there is a checkbox to put Time Machine in the menubar. From the menubar, you can start your backup anytime you like, as well as check the status of the backup.

Have you selected the disk to be backed up to? If it is not selected, then it probably remains grayed until you do.