From Mathematics to DSP
In general, Maths to audio signal processing is a very natural progression. I seem to recall my Master's and PhD supervisors both did their undergraduate degrees in Maths. 

To your questions:


This is the hardest question to answer. Depends what's in your "portfolio of basic competence". In addition to the understanding of signal processing theory that you're developing you need to show you're able to implement these algorithms probably either in hardware or software. Given little EE grounding, I'd suggest software. Look at developing your software engineering skills. They will serve you well in either academia or industry.
MSc will definitely help but is not absolutely necessary. The types of algorithms you are describing could conceivably be the subject of an MSc level project. I work at a company that makes hardware for audio signal processing. The statistic they gave me is that around 60% of staff have a Master's or higher degree.
Yes, combining signal processing and machine learning is a very wise idea if you want to work towards the research end of the field. Several problems in audio signal processing - e.g. audio source separation and speech recognition- are increasingly finding their state-of-the-art solutions come from machine learning.