This is not a question of rules that have exceptions for certain words. As with all issues of phonology, most native speakers are actually unaware of the existence of any rules. Depending on our particular accent, we either follow the rules instinctively or we don't. The only conscious decision we might make is to enunciate more carefully or emphatically in some contexts, and this can affect how we say the word 'the'. As Avi has explained above, there is a regional aspect to this. Some American accents, particularly in the south, don't follow this rule and speakers will use the 'thuh' pronunciation in all contexts. Or, as Alix says, they will reserve the 'thee' form just for emphasis. The phonological rule itself has no exceptions for particular words or sounds. If our accent demands that we change 'the' to 'thee' with one vowel sound, this happens with all vowel sounds. Note that is about vowel SOUNDS, not letters - the written spelling is irrelevant. For example, we say 'thee(y)egg' but 'thuh Euro' (because 'Euro' begins with a consonant sound). Likewise, we say 'thuh house' but 'thee(y) honour' (because 'honour' begins with a vowel sound). It's also worth noting what happens in accents which 'drop' the initial 'h' - something which is very common in many parts of the UK. In these accents, 'house' is pronounced 'ows' (with an initial vowel), so speakers say 'thee(y)ows', in line with the rule. Note also all speakers (whether or not their accent follows the the/thuh rule) will occasionally use 'thee' for emphasis before consonant sounds. For example: "Guess what? My sister's going out with Michael Jackson." "What? THE Michael Jackson?" (elongated to 'theee' for emphasis). "Don't be stupid. I mean Mike who works at the garage. His surname's Jackson."

You can use this as a guide but it is not strictly a rule: "thuh" before consonant sounds "thee" before vowel sounds Maybe they say "thuh" in Dora for everything in order not to confuse their young viewers with nuances.

This is mostly a locational/dialectic difference in English. It is generally accepted that certain areas of the United States and Canada replace "thee" with "thuh" in everyday speech but that speakers from the UK and Ireland tend not to do so. I suspect you'll find that most American speakers in particular do it almost all the time -- except in formal environments, where even the most rural speakers tend to become far more formal and adhere to the stricter rules of pronunciation. Realistically, I doubt most people would find it odd either way as long as you are consistent -- for example, nobody will think it weird if you say "thee ocean" or "thuh ocean" but if you switch sentence to sentence, it will start to attract notice.

In general you can replace any occurrence of "thee" with "the" but not vice versa. "The" is more commonly used while "thee" is used for emphasis. As you mentioned there are some cases where you can use "thee" without emphasis, but I think it's best to mostly use "the" until you get the hang of where you can use "thee".