Let me start by saying that I have read many posts on Cross Validation and it seems there is much confusion out there. My understanding of that it is simply this:
when you perform k-fold cross validation you are already making a prediction for each sample, just over 10 different models (presuming k = 10). There is no need make a prediction on the complete data, as you already have their predictions from the k different models.
What you can do is the following:
train_control<- trainControl(method="cv", number=10, savePredictions = TRUE)
Then
model<- train(resp~., data=mydat, trControl=train_control, method="rpart")
if you want to see the observed and predictions in a nice format you simply type:
model$pred
Also for the second part of your question, caret should handle all the parameter stuff. You can manually try tune parameters if you desire.