Can I start full stack development course in 7th sem ?

After completing your VI semester, please find out what so far you have learned about the software development tools and website creation fundamentals. Then go through the syllabus of the VII semester and see what are the things that would be subsequently covered there. If sufficient coverage is there then you can learn these things by the end of VII semester. If it is not so as per the expectations then I am afraid you will have to go for some extra studies and may have to join some online class for full stack development learnings. Generally such courses are a bit costlier and come with a time schedule of 300 to 350 hours of learning.

Basic knowledge of software development like Agile and Scrum methodologies, HTML, CSS, JavaScript to create dynamic websites, and GIT to manage version control systems would be required to begin the learnings for reaching to the goal of achieving full stack development. This further constitutes of learning software development and testing technologies for example like JavaScript, Node.js, Angular, Docker, and Protractor. This would then be followed by creating end to end application, test them and store the data in some data base application. Then comes the task of creating end to end websites/ scalable and dynamic websites using say Front and Back-end frameworks, REST API, Node.js, and Express.js. Some knowledge of hosting the website and porting the Applications in cloud would also be required.

Full stack development consists of learning many elements and if one is really interested in that then some hard work would be required to achieve this target. One has to study some additional elements which are outside the realm of the syllabus and that has to be seen thoroughly beforehand before joining any online activity in the matter.

That depends on your interest and performance level. One thing you have to understand is by starting this new course, do you feel that you may not be able to concentrate on your 7th sem examinations
It is more important to complete your B.Tech with good score. So you should give importance to your B.Tch. But if you feel by spending some extra time you can cope up both simultaneously you can join
You have to work hard and focus more on studies and you have to do justice for both the courses. Once you have B.Tech and additional certificates in stack development course, you will have more chances to get a good job in that line. You should start searching in various job portals and apply for the same.