How to measure biosignals with 1800's technology?
You would not use any electronic amplifier at all, because you simply were not able to build one without tubes (which only arrived at marked after 1900) or semiconductors.

What people actually did back then was to design mechenical devices, to show a visible signal amplitude. I guess the most significant one was the already by MarkU mentioned String Galvanometer. It used a very lightweight and thereby sensitive wire, which was deflectet in the elcetrical field of the signal. To make this little movement visible without influencing the signal itself (or decreasing the amplitude due to some load) the amplification was done optically by projecting the wire on the wall.

Regarding your direct tasks: These questions do not aim an the design of an "opamp" but on filter circuits and electrodes. You can use the named passiv components to build filtering circuits to remove noise components of your signal. First of all you would want a low pass filtering. The electric signal from the heart does not contain any information in the frequency range over 100 Hz - so you can reduce noise a lot by limiting this upper frequency. Additionaly the respiration will distorte the baseline of you signal with a frequency below 1 Hz - so you want to also block these low frequencies. This can be done by resistors, capacitors and inductors.

The electrodes are another topic: You can not simply use metal wire to connect to peoples skin to your apparatus, because you have a transition from electron conduction in the metal to ion conduction in biological tissue. A simple metal electrode will polarize and increase in resistance a lot. Therefore you use electrodes, which are designed for this transistion and which can deliver the needed ions. A reasonable example would be silver/silver chloride electrodes, although back in 1800 I guess you would just put the extremities in large metal buckets of saltwater.