The first day I started a history class in college, the professor said "Forget everything you ever learned about history; none of it applies here!" So I'm going to make the same suggestion about web design because PageLines Framework has dramatically altered the way sites are designed and developed. There's no need for Dreamweaver or any other third-party editors.
What you'll want to do is install Base Theme (free at Dashboard > PageLines > Store > Themes). Use that to apply most of your CSS changes and functions.
Once you complete the site design and layout, you can use the export capabilities of PageLines to download the base theme to place it on the new site. You can also use Dashboard > Tools > Export to get all the database content, etc.
I've been using PageLines products for two years now, and proudly have to say I run one of the largest sites using Framework, having more than 100,000 posts, 5,000 pages, and almost 90,000 images. It's one of the top food sites online, and has been around for 16 years.
I can't answer all of your questions, but I'll give it the ol' college try.
What is your favorite way to customize Pagelines to look how you want? Child theme?
Personally, I design directly to the Framework, but strongly recommend using the Base Theme.
How do you prototype your layout designs?
I set up a unique WP installation, and do all my design work there, then copy the CSS to the regular site when I'm satisfied with the design. I also validate the CSS before final implementation by using the Direct Input tab here: http://jigsaw.w3.org/css-validator/. This prevents CSS coding errors from messing up the site. I also use Firebug for Firefox, but if you prefer Chrome, it has an Inspection Tool built right in. This helps to test various design options available to you.
Do you develop locally in MAMP? Or on the live server?
I do it on a live server, but many use MAMP.
How do you push to the live server? FTP? GIT? Something else?
I use the export/import tools available in both PageLines and WordPress. You really shouldn't need to use FTP or anything else to migrate the site or its child theme.
Do you use Git version control in your workflow? Beanstalk App?
Never had to do this at all.
Dreamweaver? CSS Edit? Coda? Sublime Text 2? What tools do you use?
Aside from Firebug and the CSS validator, I use several websites to help me along: http://colorpicker.com/ to help me find just the right color codes; http://www.w3schools.com/css/ to help me fine-tune CSS, particularly with its testing capabilities, making it a great learning asset; http://validator.w3.org/ -- like the CSS validator, I use this when I'm adding HTML code in text widgets, letting me make sure I don't break things. Lastly, though not in order of precedence, I use the amazing tools available in the Store, such as Hooker Plugin (to apply hooks), Action Map (to find function hooks) and others. For graphics, I use Corel's Paint Shop Pro, but any good graphics program will do very well. Most people use Photoshop.
What is your optimal development workflow?
I'm not sure how to answer that. Including graphics and custom coding, I can usually create a great design layout in an hour or less.
Why do you like Pagelines over a regular Wordpress theme?
PageLines offers so many more resources, and design options, without the limitations of someone else's vision. I have complete control, speedy tools, and no restrictions about how I design the site. I can do so much with it, and increase functionality, rather than being constrained by an off-the-shelf WordPress theme.
I know this won't really answer all your questions, but hopefully, it addresses some of them.