While teaching writing, do I need to correct all the errors and let the student learn from my corrections?

The skill of writing proficiently is generally seen as a mammoth task by many students. This therefore requires sensitivity when it comes to giving feedback that will be effective in bringing about progress in  students' writing.

When correcting a piece of work, it is important that the student is not a recipient to tons of correction marks which could easily lead to demotivation and lack of confidence.Before writing tasks are given, students need to be aware of the focus of the task i.e. speech marks, capitalization, spelling , word order, descriptions etc. When marking this work give a double tick at the place where the student has met the target and an 'upside down v' where they have missed the opportunity to meet the target.  There are ofcourse tons of ways to mark or represent these.

I usually practice SID marking in students' books where I state one strength (e.g. Well done for using speech marks correctly), one need for improvement (e.g. focus on spelling)and one discussion question  where learners need to observe their work and to give their answer in writing to me (e.g. When do we use capital letters?).  In this way, the student receives balanced feedback that motivates and allows them to focus on their points of development in steps rather than being overwhelmed by their own errors.

Yes

I even choose pattern mistakes from the assignment and put them all in one extract and ask Ss to correct them

Yes, writing teacher needs to correct learners' errors to let them know where they have gone wrong and why. In communicative approach too, writing errors need to be corrected, speech errors need not.