No debug mode in TextExecute that I'm aware of. It doesn't have an editor window so wouldn't make sense really. You need the full TestComplete IDE for that.
And for cross-browser compatibility, you need to have a good look at how it capturing and mapping objects. Different browsers report different properties so you need to make sure you're using something that is consistent between all the browsers you plan to use.
I've also found that the same site reports itself differently to TestComplete. My current project has to be run using both Chrome & IE. IE reports duplicate objects inside containers which are not reported by Chrome.
Your best bet is to be VERY careful and thorough when first mapping object you plan to use in your tests and be 100% sure you are capturing and storing them in such a way that they'll work in both before you write any code round them. One thing I've learned about using object/GUI maps is it's a bit like decorating your house .... it's all in the prep. If your map is good, writing code that uses it without problems is way, way, easier ...
Have you tried having a look at some of the problem objects using the object spy to see what the differences are in how they're reported?
