Greene not only are figs safe for dogs they actually love them. I give my dogs the ones the birds pecked or yellow jackets got a hold of all the time. Dormant cuttings are the easiest to root and during dormancy is the best time to do any major branch removals on fig trees and any other trees for that matter. Once pruned in winter you can do some light early summer pruning in following years to control height. Too much pruning during the growing season will encourage new growth that is easily winter killed even in zone 8b.
If you are thinning your plants branches to allow air circulation in hopes to control the rust just know it won't cure it but air circulation is good for the tree regardless so by all means do open it up a little to allow not only air but light to penetrate also. This will help dry any damp leaves and help control the spread of rust but again won't fix the problem. I'm sorry to hear about your rust issues, in the humid south there's no escaping it unless you are willing to use a spray regimen on a regular basis. Trees can go 5, 10, or even 30 years without rust and then boom one year there it is. It' much more than just an eye sore it will defoliate your tree and the remaining figs will not ripen. Not only that but defoliation will also encourage new growth that is not winter hardy resulting in winter die back also. Even temps of only 28F can damage new growth resulting in a loss of your breba crop aka early crop "the figs grown on the previous years growth". You can still get a main crop BUT and it's a big but if the rust is not controlled those main crop figs will not ripen before leaf drop happens next fall and the process just continues to repeat it's self year after year.
If spraying is not an option for you LSU purple is often referred to as rust resistant and varieties like LSU gold & Desert King often ripen before leaf drop occurs so these may be good options for you. I'm not a fan of LSU purple it's flavor is kind of blah even more so than brown turkey but LSU gold is one of my favorite yellow figs.
