Just double checking myself on something here. Monster's can move twice, right?

Can only attack once, but can move twice if the only viable option in their action tree is engage, and then engage again if they still aren't adjacent to a hero (and no other options are doable)?

Yep, they can move twice (minus zombies) and attack one time ( minus monsters with ravage skill or other app effect). If you dont get a valid option in the app, you can circle the options until you get a valid choice.

Example: 1. Attack the adjacent hero 2. Engage 3.use a skill in the adjacent hero

No adjacent hero, use step 2., if no adjacent hero yet, you skip the step 3 and step 1, and use step 2. again.