Q3. Embedding codimension (sometimes simply codimension).
Q1. Edit: Yes when $R$ is complete: choose a regular local ring $S$ with the same embedding dimension of $R$ such that $R=S/I$ and localize. I don't know in general.
Q2. For excellent rings. An indirect proof, but short enough to be written here, is as follows. Embedding dimension is the diference between the invariant $\delta_2$ in "Ragusa, On Opennes ..." which is upper semicontinuous by Proposition 3.6 in that paper, and the complete intersection defect which is upper semicontinuous by the main theorem in "Alonso-Rodicio, On the upper ...".