Thanks for posting, and welcome Fusion360!
I think I can answer your questions.
1) How to join end Points of two separate lines/T-spline lines?
I can answer this for sketch lines: Use the Coincident constraint, and select the endpoints of the two lines, or any sketch curve. You can do this by either pre-selecting the two endpoints and choosing Coincident, or by selecting Coincident first, then selecting the endpoints after. Here is a screencast of this: http://autode.sk/1PdqMP7
I'm not sure exactly what you mean by joining the endpoints of T-spline lines. Can you show an example?
2) How to extend or continue drawing from a line end Point?
There is an Extend command in sketch: http://autode.sk/1CsYBBA
To continue drawing a new line, just pick the line end point as the start point of a new line: http://autode.sk/1aE0HZW
3) How to duplicate a sketch to another sketch layer?
I assume that what you really want to do here is to copy geometry from one sketch to another sketch. Copy/Paste can do that: http://autode.sk/1Pds8cF
Hope this helps you make progress!
Jeff Strater (Fusion development)
Jeff Strater
Engineering Director