1) Questions of terminology like that are always debatable! I would say that the term "analysis" describes techniques to convert the speech signal into another representation, closer to the semantic level than a raw list of samples (such as a written transcription in the case of speech recognition, a visualization of the tone curve in the case of a system assisting Mandarin learner, the identity of the speaker in case of speaker identification system, a category of emotion in case of an emotion detection system) ; while "processing" describes problems in which both the input and the output are speech (speech denoising/enhancement, coding and decoding, source separation, 'creative' effects like pitch/timing modifications). Of course, there are contexts in which "processing" describes the whole field, analysis included.
2) Deterministic signal processing (filtering, transforms), statistical signal processing (auto-regressive models, sinusoidal model estimation), statistical modeling and machine learning, psychoacoustics, knowledge of how humans produce speech... If you venture into recognition: language theory (finite state machines), natural language processing, some elements of distributed computing (for large model training), data structures/algorithms. If you venture into speech coding or enhancement applications: information theory, experiment design (for subjective evaluation campaign), DSPs/embedded systems with hardware acceleration or even in some cases hardware design. Tools and language: mostly matlab or scipy for research, mostly C/C++ for implementation; and a scripting language (perl, python) is always helpful when you have to deal with texts/annotations.