Does the model(best fitting line/curve) changes when the training data is changed in the cross validation?
Your intuition that the model changes depending on the data input is right. What changes is namely the parameters, e.g. weights if you are performing a regression analysis.
Weights change primarily when you are training the data. You can reliably say that whenever you train the model it changes (namely its parameters) .
During cross validation, you split your datasets into an arbitrary number of subsets. You use some of them as training sets, while you use the rest as validation sets, and measure the error.
You repeat this through and through and you measure error of each. In each iteration when you choose different subsets (kth iteration uses different test-validation sets than the k+1th ones), you do change the model as you are re-training, (however this should not be important as you are doing this to have an approximation on how the model will perform with new data). Finally, you'll have an estimation on the model's error, (but you can use this process to choose features with most predictive power, if you are performing cross validation for feature selection). Edit: It is however important noting that cross validation is used primarily for getting an estimation on how the model would perform in general, not for training test selection.
This goes for numerical models, but I do think they apply to decision trees and other probabilistic models as well, because this process would change information metrics that are given to the model as an input (e.g. split criteria of the decision tree).