RE: The puzzling thing is why would it work consistently on one page but not another.
??This is not unusual. Different JavaScript will typically be loaded on different pages depending on the content. You can use this as a clue to find the problem JavaScript.
RE: The theme developers say they can’t provide support for third-party plugins.
If you are correct and the problem is related to your theme, this is a cop out and a warning sign of a poorly supported theme IMHO. If a plugin or theme conflicts with CF7, it probably also conflicts with many other plugins. It’s generally a sign of a poorly coded theme or plugin and one that people should avoid using.
???RE: How would I go about figuring out where in the theme the issue is?
You could run a JavaScript trace in Firebug to see where the JavaScript stops responding – see https://getfirebug.com/logging??.
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RE: Anyone else having a similar issue?
This is a very poor question to ask in this case. ???With over 23 million downloads it’s inevitable that other CF7 users will have similar problems each day due to JavaScript conflicts. But their problems will be due to their particular WP installation configuration which is almost certainly entirely different to yours. As such their comments are likely to confuse things rather than help you solve your particular problems.