Haven't seen those errors before, could be due to a web server resource limit or setting perhaps. How many minisites are you using? I understand you can't save changes to minisites, but can you save the overall theme options by clicking the blue button within the theme options panel?
Sounds like an issue with max_input_vars limit especially since it changed going from one host to another (because the server setting could be different and/or PHP version could be slightly different). Check the solution here: http://industrialthemes.ticksy.com/faq/1055
Please see this FAQ with the solution: http://industrialthemes.ticksy.com/faq/1487