Electronic – Amplifier output goes haywire with load attached

The book I learned from, and that I still see new hires bringing with them from school was Grey & Meyer (now Grey, et al.), Analysis and Design of Analog Integrated Circuits. This was the text in the main analog design course at Berkeley in the '90's, and I believe it's also used at numerous other institutions. It's got as much theory as they teach undergrads anywhere in the country. Of the topics you expressed specific interest in, this book does cover current mirrors, differential stages, and feedback theory. It also has a very quick chapter on device models of diodes, BJTs and MOSFETs. It doesn't go particularly deeply into device physics -- only enough to motivate the models.

On the other hand, its heavily aimed at future IC designers, and at CMOS op-amp designers, which implies a whole different set of design techniques than you'll use building analog circuits from discrete components or op-amp ICs.

For board-level circuit design, app notes from the various vendors are probably as good as any textbook. For example, TI's Op Amps for Everyone, or Analog's Op Amp Applications Handbook. Even if these guides will tend to recommend a particular vendor's products, they do provide a strong background in theory that applies to using any vendor's parts, and they do provide a good balance between theory and practical issues. You haven't stated what you mean by "wideband" applications, but op amps are available off the shelf with gain-bandwidth products above 1 GHz, so these op-amp-focused books may apply better than you think.

It is not an schematic problem. It is implementation problem. What type of prototyping you are using? IMO, this schematic have to be prototyped carefully on a PCB, not on breadboards and any kind of rat-nest type of solder-less technologies.

Note, that such high gain, precision schematics are highly sensitive to all kind of EMI and must be constructed correspondingly.

Take special attention to the input circuits - make them with screened cables that are properly grounded and as short as possible.

Yes, the instrumental amplifiers suppress very efficiently the common mode EMIs but on gain 750 and bad construction, some differential EMIs can be inducted as well.

Edit (after PCB images has been posted): This PCB is not well designed. Routing the power and ground as such a thin tracks is wrong. You must use wide copper areas for the ground and tracks as wide as possible for the power tracks. Your PCB must be almost full of copper. Also the density of the PCB seems to be a little bit small. Higher part density will short the tracks and lower the possible influences.