Ok, let's break down the questions:
Can I launch an app under self employment or I would need a company for that?
You can develop as a self-employed person. Ensure that before launch day, you notify HMRC of your self-employment so that you do not find yourself in court for tax evasion.
Can I copyright the app under self employment?
Yes. This would be your Intellectual Property. You would be the rights holder.
Has anybody has any experience with launching their application from this point of view ?
I've sold something previously that I had no intention of continuing to develop. I did this as a Ltd company.
The purpose of a Ltd company is to protect you. If you make an app that bricks every phone it touches, and you are a sole-trader (self-employed), then you can be sued personally. A Ltd company shields you from this.
Also, if you're going to make more than £50k in a year, a Ltd company will allow you to reduce your tax more creatively (while still legal).
You should have a chat with both an accountant and a solicitor (that specialises in IP, as per joeqwerty)