Applying for an Exit Visa
We can thus assume that Yaakov fears that his departure to Egypt may be against God's will or possibly even a threat to his "bechira." Therefore, Yaakov first stops in Beer Sheva to pray - to 'ask permission' to leave Eretz Canaan.
Now, to complete our answer to question (a), we must explain why he stops specifically at Beer Sheva. The commentators offer several explanations:
Although each commentator quotes different sources to explain why specifically Beer Sheva is chosen, all agree that Yaakov is indeed worried about his departure from Eretz Canaan.
This very same concern of Yaakov answers question (b), as well. Yaakov prays specifically to the God of Yitzchak because when Yitzchak faced a similar situation, God did not allow him to leave Eretz Canaan. It is Yaakov's hope that despite this precedent, God will permit him to leave. [See Radak and Seforno.]
[Note that Ramban offers a different approach (based on what he calls "sod"). Ramban explains that Yaakov recognizes that his departure to Egypt marks the beginning of the long historical process of Yetziat Mitzrayim. Realizing that this process may entail terrible suffering of "midat ha'din," Yaakov prays specifically to "pachad Yitzchak," the manifestation of God's providence through "midat ha'din," in hope that his children will suffer as little as possible.]