Tons of problems trying to simulate simple rope without elasticity

[josemendez]

rope and paper clip NEVER stops moving while the pencil is static. Here is the video.

Chances are the rope and the clip are colliding with each other. This will result in jittering, attachment offsets and all sorts of weird behavior as described in the "Pin constraints and collisions" section of the manual:

http://obi.virtualmethodstudio.com/manua...collisions

Quoting it:

When pinning a particle very close to or inside a collider (so that they overlap), if you have collision constraints enabled you can encounter a situation in which both constraints fight each other. This results in jittering and/or an undesired offset in the pin position, because the collision and pin constraint cannot be met simultaneously. If the particle is pinned to a rigidbody, results will be even worse as this setup causes a force feedback loop with largely undefined behavior.

Just filter out collisions between the rope and the clip using collision filters, as suggested in the above link.

rope behaves like a fishing string moving the pencil up and down. Here is the video. 
I need the rope similar to this  As you understand such type a rope don't stretch at all an don't.

Crank up the amount of substeps (setting found in the ObiFixedUpdater). That will do. The manual contains a very detailed explanation of how and why the timestep and the amount of iterations affect the perceived elasticity of the simulation, and why using a smaller timestep is more effective:

http://obi.virtualmethodstudio.com/manua...gence.html

Also keep in mind that the higher your rope blueprint's resolution, the more substeps you'll need to remove any spurious compliance (since the rope contains more particles and constraints). It's usually a good idea to keep resolution around 0.25-0.5 unless you really need full resolution, this is specially the case for long ropes.

Furthermore it looks like paperclip has 10kg of mass while it's 0.01 only.

The mass of the paperclip doesn't matter by itself. What's important is the mass ratio between the clip and the rope. If your clip mass is 0.01 and your rope is 0.001, then for all intents and purposes your paperclip weights 10 times more than the rope. You'll get the same results you would with a 10 kg clip attached to a 1kg rope.

the worst issue - if I move down the pencil and paperclip collides with a surface, it behaves veeery weird - instead of laying down to the surface it "floats" in the air, and the rope starts to shake as crazy. Here is the video.

Your clip and the rope are colliding with each other, as I pointed out above. The situation is this: the rope is attached inside a collider, so the attachment wants the rope to be inside the collider. However the collider wants the rope to get outside. Since the rope can't be simultaneously inside and outside the collider (for obvious reasons, you cannot be both inside and outside someplace), jitter and weirdness ensues as the engine tries to solve something that's impossible to solve.

The simplest solution is to deactivate collisions between the end of the rope and the clip, as proposed in the manual.

artifacts with attached objects - looks like the attached paper clip (but I also will add more attachments) position doesn't follow the particles position properly. You can see in the video I mentioned in #3 there are some gaps between paperclip pivot point and end of the rope

Same issue as above: there's a gap because the rope is colliding with the clip.

I also checked all the test scenes, and still can't get, why it's so complicated to implement it.

Deformable body physics is really complex, advanced stuff. Most people struggle understanding regular rigidbodies well enough, for deformables crank up difficulty a couple notches. Getting comfortable with it will take some time, so hang in there (no pun intended ). Let me know if I can be of help.