Thanks for your response,

1. Yes
2. p(y|X,w) is likelihood distribution and defined as Bernoulli for logistic regression, so it is nothing but the pmf of Bernoulli
3. Each instance has its own probability of success and it is defined by sigmoid of linear mapping which differs from instance to instance. All instances share the same weight w and it is model parameter that we want to learn but x(t) and y(t) are depend on t and t indicates which instance we are using. So when x(t) change, probability of success also changes.