What are safe characters to use in SQL Server Full Text Search?
It is difficult to answer the question specifically without a good description of what you are trying to achieve and why, but it appears that Full-Text Search may not be a good fit for you based on your need to search for characters that are either reserved for a special purpose within the full-text engine, treated as delimiters by word-breakers, or present as a stopword in the default system stoplist.

The Full-Text Search feature is primarily aimed at performing linguistic searches on words and phrases using the rules of a particular language like English, German, or French, for example. Searching for programming keywords like $partition in T-SQL will probably not work as you would hope because $partition is not a word or phrase in any of the available 50 (spoken) languages. If there were a T-SQL language option, things would be different.

It might be possible use a custom dictionary of 'new words' to meet your requirements, but this is not something I have directly experimented with myself. I think that's about as helpful an answer as I can offer without knowing more about your needs.