Tokenizing words of length 1, what would happen if I do topic modeling?
The libraries usually exclude 1-length tokens and tokens with no alpha-numeric characters because typically they are noise and do not have any descriptive power. That is, these tokens are usually not helpful, say, in distinguishing between relevant vs not relevant documents. 

However, if in your domain you feel like 1-length tokens can be helpful, feel free to use them as well. For example, if all the document that contain 1 belong to the same topic, it may be a good idea to preserve this token. 1 has descriptive power in this case: it can help distinguish between one particular topic and the rest.

Now, your next question is about LSI. For LSI there is no difference if a column in the document-term matrix corresponds to a 1-char token or to a 5-char token. So you can use LSI in your analysis.