OSX is kinda special when it comes to java since java is kinda required by the base OS. Here is a link which provides information on how to run multiple versions of java on OSX:
Modpacks tend to take a bit for them to show up on the launcher, I would suggest waiting 24 hours.
Looks like you are attempting to load a 1.7.10 mod
[02:43:20] [main/TRACE] [FML/]: Found FMLCorePluginContainsFMLMod marker in 1.7.10-MB_Battlegear2-Bullseye-188.8.131.52.jar, it will be examined later for regular @Mod instances
On a 1.10.2 server
[02:43:20] [main/INFO] [FML/]: Forge Mod Loader version 184.108.40.2063 for Minecraft 1.10.2 loading
You need to upgrade java. According to your crash log
Java Version: 1.6.0_65, Apple Inc.
Java VM Version: Java HotSpot(TM) 64-Bit Server VM (mixed mode), Apple Inc.
You are running 1.6, you generally need 1.8+.
Make sure your file is flagged as either Beta or Release otherwise the entire project will not show up.