Here are the answers:
1. In sequence modeling, we expect a sentence to be ordered sequence, thus we cannot take random words (unlike bag of words, where we are just bothered about the words and not really the order). For example: In bag of words: "I ate ice-cream" and "ice-cream ate I" are same, while this is not true for the models that treat entire sentence as ordered sequence. Thus, you cannot pick N random words in a random order.
2. Choosing tokens is model dependent. You can always preprocess to remove stop words and other contents such as symbols, numbers, etc if it acts as noise than the information.
3. I would like to clarify that lemmatizing and word-piece tokenization is not the same. For example, in lemmatization "playing" and "played" are lemmatized to "play". But in case of word-piece tokenization it's likely split into "play"+"##ing" or "play"+"ed", depending on the vocabulary. Thus, there is more information preserved.
4. max_length should be optimally chosen such that most of you sentences are fully considered. (i.e, most of the sentences should be shorter than max_length after tokenization). There are some models which considers complete sequence length. 242
Example: Universal Sentence Encoder(USE), Transformer-XL, etc. However, note that you can also use higher batch size with smaller max_length, which makes the training/fine-tuning faster and sometime produces better results.
5. The pretrained model is trained with MAX_LEN of 512. It's a model's limitation.
6. In specific to BERT,as claimed by the paper, for classification embeddings of [CLS] token is sufficient. Since, its attention based model, the [CLS] token would capture the composition of the entire sentence, thus sufficient. However, you can also average the embeddings of all the tokens. I have tried both, in most of my works, the of average of all word-piece tokens has yielded higher performance. Also, some work's even suggests you to take average of embeddings from the last 4 layers. It is merely a design choice.
7. Using sentence embeddings are generally okay. But, you need to verify with the literature. There can always be a better technique. Also, there are models specific to sentence embeddings (USE is one such model), you can check them out. 716