Word embeddings (or subword embeddings) are used for nearly all recent NLP algorithms, both shallow (e.g. FastText) and deep (e.g. Google Neural Translation). Unless you're using a basic bag-of-words approach, you need to translate your words into some vector format, so you probably want some kind of embeddings. In practice, all the state of the art approaches for translation, language modeling, classification (eg sentiment analysis), etc. all sit on top of embeddings.

It's not word embeddings job to handle phrases - but nearly all modern phrase embedding algorithms sit on top of word embeddings. They often create a weighted average of embeddings by using an attention model, or they can use a more complex model such as an LSTM with attention (e.g. CoVE - https://arxiv.org/abs/1708.00107 ).

Word embeddings can handle polysemy - high dimensional vectors can (and do) hold information of various types that is used in different contexts in different ways. Some approaches deal with this more directly (e.g. including part-of-speech as part of the vocab item), and that sometimes can help a bit.