I figured out the issue. The TBB was missing parameter schema "Generate dynamic component presentation". Once I added that, I got the required values in "LinkedComponentValues"
Never mind, the issue was with recent changes to event system code.
Our issue got resolved. We had some internal URL Rewrites which was causing the issue. Corrected it and "Update Preview" started working in IE too.
I found the issue and fixed it. Tridion\web\WebUI\Models\SiteEdit\Web.config had below line commented. Uncommented it restarted tridion services and issue got resolved.