Word Embedding Interpretation
Yes and yes. So, if you have "I" [4.55, 6.78], "like" [3.12, 8.17], and "dogs" [1.87, 10.95], each embedded representation roughly equates directly to each word, and thus the order isn't lost when the embedding is done. And yes, the shape would be (batch_size, 600, 15) for batches of 600-word-sentences and embedding dimension 15. I think the question you're indirectly asking is something like "Does each word directly correlate to a single embedding vector of length embedding_dimension?" aka "Does 'I' directly correlate to [4.55, 6.78] independent of the other words/embedding vectors?" For the most part, the answer is yes.
For what it's worth, it's been useful to think of it like languages. Doing a hash representation (excluding the duplicate values) or a categorical column with a unique value for each word is somewhat akin to how classic Chinese is, with a unique character for every word. Whereas embedded representations are more akin to the English language, with a "word" being a fixed number (embedding dimension) of letters (floats). The advantage is similar to how we gain advantages in the English language. For example "dog" vs "dogs" has 3 similar characters because they are very related concepts. Similarly, you can take advantage of embedding by representing "dog" as [1.23 4.56 7.89 1.12] and "dogs" as [1.23 4.56 7.89 9.87] or some such.
Random but I hope this helped. Good luck~~ =)
Before I ask the question, let me preface this by stating that this question has been answered in many articles, but I still struggle to understand the basic format of word embeddings. 
Let's start with the sentence "I like dogs". Assuming a simple hashing approach, "I like dogs" can be represented in the vector [1, 4, 6] where the elements of the vector correspond to the hash of each word (assuming these aren't the only words in the vocabulary). From what I understand, this vector is fed into an embedding layer which adds an extra embedding dimension onto the input tensor of the RNN (doesn't have to be vanilla RNN).
The embedding tensor (with lets say an embedding dimension of 2) will look something like this for a single entry in the batch:
[[4.55, 6.78], -> I 
[3.12, 8.17],  -> like
[1.87, 10.95]] -> dogs
This tensor has the shape (1, 3, 2). Does the length of the second axis (3 in this case) always equal the length of the input vector and therefore represent each individual word in the sequence or do I have a fundamental misconception of how the tensorflow embeddings work?
To clarify: say I had a much longer sentence with 600 words, would each word after embedding be remembered in their original order and be represented by a vector of whatever size was chosen for the embedding dimension (we'll say 15), thus making the shape of the embedded tensor (batch_size, 600, 15)?
Note: these are just random numbers and don't represent anything in particular.