"like I" or "like me"?

The rules you were taught are artificial. It is very rare to hear "as I" used in the way you have it in your examples out in the wild. It may be correct according to prescriptive English grammar, but it is not idiomatic to the language until you add the extra bits. One would say either:

She is smarter than me.

or

She is smarter than I am.

The same pattern emerges when looking at like. Used by itself, in idiomatic English, you would use the object pronoun when the word is used alone, and the subject pronoun when the phrase extends into a sentence-like structure.

Despite the vain longings of those few people who want English to be nice and neat (and have a one-to-one correspondence with Latin and Greek), our language has its wrinkles and inconsistencies. This is one of them.

As with many prescriptive "rules", the rule you mention (a) generally does not reflect the actual usage of educated native speakers, and (b) is based on spurious argumentation.

The argument is usually based on a couple of (probably largely unfounded) presumptions, such as:

• that "He's as tall as [me/I]" somehow has to be "short" for "He's as tall as [I am]"
• that sentences like "He's as tall as me" somehow introduce some ambiguity that can't be resolved by context/other natural means.

The spuriousness of the first suggestion can be shown, for example, if you look cross-linguistically. In French, "plus/aussi grand que *je" is not grammatical, and no French speaker is suggesting that the grammatical version "plus/aussi grand que [moi]" is short for "plus/aussi grand que [je (ne le) suis]".

In the second case, did your teacher or does your grammar book actually point to a documented case where a real-life misunderstanding occurred because of "as/than me" being used where it "should" have been "as/than I"-- and if it does, does it explain why saying "as/than I", rather than "as/than I am" (etc) is the best solution?