I will try to answer your confusion but not the individual "how to" i.e. How a Decision Tree calculates probability etc. You may search the internet e.g. "How Decision Tree works etc."
Score is mostly used to describe the over-all prediction result of the model. There are many different techniques i.e. metrics for this e.g. Accuracy, RMS, ROCAUC depending upon the prediction type and some other specific conditions.
Probability tells us how confident the model is about its prediction.
Different models have different techniques to calculate the probability but in a very simple language, its proportional to the distance of the data points from the decision boundary the model has learnt from the training data.
As in the image, upper red-circled will have more probability of being "Yellow"
is it fair to say that for a score 10 the instance is more likely to be successfully predicted than a score 5
Yes, you can see that, the lower red circled point has more chance of being miss classified due to model bias.