Do widgets slow down Android phone? I use a lot of widgets (like Gmail, calender, weather app, Facebook, music, news, etc.) and since I have been using them I feel my mobile has become slower than before.

I don't use a task manager. Watchdog alerts me on Gmail only. My RAM is 512MB.

Are there any studies that reflect Android performance with the use of widgets?
I share my experience and knowledge as an android developer:

Most widgets don't consume (or use just a bit) processor power when they are not visible, but some would consume even if your screen is off and phone is locked. It is based on what task the widget is considered to do, and how developer has written and optimized the code.

As a general rule: more widgets would slow down your launcher (desktop) because your phone has to render UI of the widget every time it is shown to you.
As I see, your phone is a low-end one and surely, it shows some lag when you add more widgets, but keep in mind that smartphones are made to do TASKS, not to just scroll fast and be lag-free. So, if you really use those widgets, keep using them!

And one more thing, if your phone is running on Android ICS 4.0 or newer, you wouldn't need any task manager or task killer. Android itself best manages tasks and you shouldn't worry about background tasks, as long as you haven't installed any malicious app.
killing background apps NEVER improves speed on your phone or battery life.