In this case you can definitely hear the "ה" sound.
However I guess it's more of a question of style, some people like to pronounce the "ה" sound (me included), while others don't. Probably not pronouncing it is more common today. Oh, and yes they will both sound the same in that case.
Both עם and את mean "with" and both of them are inflected, so, "with me" can be both איתי or עימי. However, nowadays, את with the meaning "with" is outdated and not used, but its inflected forms are used - איתי for example. On the other hand, the form עם is used, but its inflected forms, such as עימי, are considered very formal and not commonly used.
Since some (many?) people still seem unaware of the Tips and Notes’ existence: those for this specific unit do tackle this point (and confirm danny912421’s information).
Notice that some prepositions have different stand-alone forms. A prime example is עם (with) which turns into -אית when suffixes are added:
with the dog - עם הכלב
with him - איתו
The reason for this is that originally we had עם (along with עימי, עימך, עימנו etc. which have since become rare) and את (et), functioning as "with", not as the direct object marker. Nowadays, את is only ever used as the direct object marker.
And here is a link to the full compilation of Tips and Notes for the English-speaking Hebrew course.
For me, as o non native English speaker, I would say that - women is refering to a more general description of the female human being - ladies is refering to the more "fancy, playfull" aspects of that same female human being. At least, this is the perception, as a man, I have of those two words... Women surely may have a different feeling about this. Not at all trying to diminish the value of women, as all human beings are created with the same value, but not equal.