Nathan, true, makes total sense. Upvoted.

Plus Illustrator can do this only for straight segments, but can’t extend a curved segment...
At the moment this is possible only with the Extend Path plugin from Astute Graphics. They have several modes to extend open paths, and allow to process several of them at once.
Here is an article about it: https://docs.astutegraphics.com/action-tags/extend#extend-path-tool-operation

I don’t imply you have to use it (it costs money), and Ai should improve it on its own, but if you deal with these often, this can tool be a salvation.

Currently the line extension smart guide only works for single points. Would love to be able to extend multiple points that are parallel, or even lines that are not parallel but would still snap to their own extension angle.

Natively — only manually, using smart guides and Direct selection tool.
There is a paid plugin tool, Extend Path, by Astute Graphics — it allow you to extend paths, one or many at once, by dragging: https://www.youtube.com/watch?v=zlriFIrSoDU

How to joint the anchor point with the line path?, without just drag and joint with direct selection tool. Is there any script availble to touch all disconnected node with path. Could you please give solution?

There is also a script, which can help if you don’t have an AstuteGraphics subscription:
https://gist.github.com/S4K4K0/62896e92287b40fded0d7f1614082d22
Less powerful than ExtendPath tool form the links above, but free.

Ah, now I see.
Yep, Illustrator can’t do that form the box.
There is, however a plugin (a paid one), Extend Path from AstuteGraphics, that allow to extend a path (in different modes) until it meets another one.

This path, however, won't get glued magically in a single shape, you’d have to group them both or make a compound path...

So there is also a request about making Ai to actually glue objects. It's harder than it seems (like what will happen if you change the curve another path is glued to? how should Ai calculate a new glue point and how should it deform the path that is glued?)
Vote for it here:
https://illustrator.uservoice.com/forums/333657/suggestions/44260098

Thank you for your response, and sorry for not getting back to you sooner.

Chris, I dont' get it, sadly. There is a way to close path with Ctrl+J, but I assume it's not waht you want. Can you draw a pic to clarify your request, please?

I am a student at university, and I need to be able to join end points to paths without having the enter the live paint. This would give higher editing capability, and allows me to keep the stylized stroke options like calligraphy.

I use the VectorScribe plugin from Astute Graphics. It has a tool for extending/reducing path length and it is really, really useful! My version of this plugin is now incompatible and I need to buy the new one. I don't think this is right. Illustrator should build-in these tools themselves. What are we paying a subscription for if we then have to pay more for extra tools to cover Illustrators inadequacies? The plugin also has a tool for eliminating unnecessary points which I use regularly. I am not going to pay for another plugin so I have to stick with Illustrator 2015 which renders my subscription completely pointless. ☹️☹️☹️☹️☹️