Apparent extension of magnets
This problem is best thought of in terms of magnetic circuits, which are mathematically analogous to electrical circuits with current replaced by magnetic flux, voltage (or electromotive force) replaced by magnetomotive force (MMF) and electrical resistance replaced by magnetic reluctance.
The steel that the paper clips are made of is a magnetic material, characterized by high permeability. This means that the paper clips have low reluctance and allow magnetic flux to easily "flow" through, similar to wires in electrical circuits. The "source" that powers the magnetic circuit in this case is the magnet: you can think of it like a voltage source in electrical terms. The magnetic circuit can be simplified to look like the following "reluctance ladder", with the permanent magnet connected between points A and D, where A is the contact between the magnet and the first paper clip, and D is the opposite pole of the magnet.
(image from this question)
Each horizontal reluctance models a paper clip. Unlike typical electrical circuits, magnetic circuits are "leaky": magnetic flux doesn't remain confined to the paper clips. Magnetic field lines leak out and return to the magnet through the air. This is modeled by the vertical reluctances in the figure.
As you can imagine, as you add more paper clips to the chain, the magnetic flux in the last paper clips gets lower because of leakage flux. The force on the last paper clip is related to the flux in the last link. More accurately, the force is related to how much the total magnetic energy (outside the permanent magnet) increases as you add another paper clip. This energy is analogous to electrical power dissipated in the ladder. The longer the chain, the lower the change in energy and the lower the force on the last paper clip.
How many paper clips you can chain depends crucially on the magnet, which is the source of MMF. It depends both on its geometry and its magnetization curve, which is influenced by the material. Coercivity is one of the important parameters in this case: it determines the maximum MMF in high-reluctance magnetic circuits like this.
(magnetization curve example taken from here)
When you approach the first paper clip with an opposite pole, you are providing a low reluctance return path for the magnetic flux, effectively shorting everything that comes after the first paper clip. Moreover, this second magnet can force magnetic flux with opposite direction to flow in the other paper clips via leakage paths, partially cancelling the flux due to the first magnet. Thus these paper clips no longer influence the total magnetic energy significantly, and therefore aren't pulled by a strong force.