Is it wrong to develop features using the target variable?
Not necessarily. It is called "target encoding" or "Mean encoding" and can be very useful. In your case you could, for example, use the
DfS of your train data to calculate a median value per street. But you need to carefully design the target encoding to avoid overfitting (there are different strategies to do that - see below link). And for the test data you can only use the target encoding based on your train data.
The Coursera course "How to Win a Data Science Competition: Learn from Top Kagglers" has great content on target/mean encoding to be found here.
Is it wrong to do feature engineering on the full dataset?
Not necessarily. As pointed out in Nicolas' answer you need to be careful to not leak data though.
Here's an example where it would be ok: let's assume one of your features is
date of enlisting which is the date when the property was published for sale. You could, for example, add a feature to the whole dataset called
days since enlisting which simply calculates the days between now and when the property was published for sale. However, your median is an example which results in data leakage since it is not "per row" data engineering but "across rows" data engineering applied to train and test data.
That's why the safer approach is to first split the data, remove the target variable from the val/test data and then do feature engineering. Thereby, you avoid any unintended data leakage.