It depends on the definition of accurate model, but in general the answer to your question 1) is No.
Regarding your second question (based on results in the paper of Niculescu-Mizil & Caruana linked below):
- boosted trees and stumps - NO
- Naive Bayes - NO
- SVM - NO
- bagged trees - YES
- neural nets - YES
You can test whether it is the case for your particular model and dataset by looking at the so called reliability plot:
- Create N bins based on the model output (e.g. 10-20)
- Create a scatter plot with average model output for each bin along X axis and average true probability for each bin along Y axis
Ideally, your X-Y points should lie near the diagonal Y=X, otherwise the output of your classifier cannot be interpreted as a probability of an event.
However, not all is lost and if needed, one can try to modify (calibrate) the output of the model in a such way that it better reflects the observed probability. In order to assess whether the calibration exercise was successful one might look at the reliability plot based on the calibrated model output (instead of using raw model output).
Two most widely used techniques for the calibration of classifier outputs are Platt scaling and isotonic regression, see the links below.
Note, that it is not advisable to cailbrate the classifier using the training dataset (you might need to reserve a separate subset of your data for the purpose of calibration).
Some relevant links
Predicting Good Probabilities With Supervised Learning
CALIBRATING CLASSIFIER PROBABILTIES
Classifier calibration with Platt's scaling and isotonic regression