I only use Mavericks for testing. On my Mavericks, FrostWire 4.21.8 & 5 won't open. From memory of couple months ago, FW5 crashes on startup. FW4.21 vanishes when trying to open. FW 4.20.9 opens fine however & connected quickly after I used the latest connection installer, turbo-charged after a minute. FW was stubborn to close however.
(I'll need to fix the launch option in the FW connection installer as this launched WS instead. I guess I'll revert to the previous technique i used from 2 years ago. Can't remember why I changed it, but seems to fix the issue. Multiple copies of FW on system was probably the reason. Mavericks won't let you move apps outside the app folder once it's actually there. Edit: created a script that only opens FW4, but can take notable amount of time to find right version if multiple internal HD's like mine; over 10 seconds. It refuses to open FW5.)

WS opens & connects quickly & is using Java 6. Both java 6 & 7 are installed on my Mavericks.

WireShare is listed as Java in the firewall. So check to see if your firewall(s) have java listed. I forgot about this aspect. Console also lists WS as java. (BTW Console will also list Phex as java.)

It might be an idea to reinstall Apple's Java 6 (1.6). Your original java installation might be corrupted. I know it's a large installation file but this is at least something you can try. A quick download link to Apple's OSX java 6 here.

I'm not sure why you are getting multiple error windows. I have seen this before but only after either LPE/WS ran out of memory overnight after a several day session. Normally you should only get a single error window popup, at least initially.

I think there was something else I was going to say but my computer crashed as I was replying to your posts (after a 2 week run & as I was launching Mountain Lion.)

We've at least ascertained the problem is only your MM computer. So something with the system is not liking java applications &/or not allowing them to connect.