Relationship of amplifier sound to transformer quality?

"if the transformer is hand wounded using high quality silver wires with iron-nickel alloy in Double-C configuration, will this give deep bass without compromising the highs and still sound quick & agile?"

I imagine that combination of metals would work rather well since, you know, iron-nickel alloy is mu metal. I suspect it would sound like a million dollars.

I had several encounters with the late great James Bongiorno (Hadley, S. A. E., Great American Sound, SUMO and Spectra AmpZilla fame) who would agree that transformer design and quality has a significant effect on the sound of an amp.  His very own amp designs testify to that. Look at the old Ampzilla, Son of and Grandson amps, all having huge high quality transformers. 

He told me that the transformers were one reason that old Sansui receivers and amps sounded so good. At the time I didn't agree with him as I wasn't a huge Sansui, GAS, or SAE fan. My how things change.

I came to realize and appreciate what genius he was, and that more often then not everything he said was true.  All of his designs are musical and engaging as was the man.

N 

Two tube power amp designers/builders who say yes are Tim de Paravicini (EAR-Yoshino) and Roger Modjeski (Music Reference). Roger will personally hand wind you one for $1,000. Tim says he can tell you how an amp reproduces bass by looking at the transformer.

Another couple designers (one already mentioned) who think alike are Kondo Audio Note Japan & Convergent Audio (CAT). 
For tube amps the bigger the xformer the better for its bass. In the CAT JL2 the output xformers are 55lbs each (which contributes to the total 180 lbs) & CAT also winds his own xformers.
In s.s. amps the quality & size of the power xformer is important. Shielding the power xformer to attenuate the noise has a big effect on the amp's micro-details. Plitron makes some very good low-noise toroidal xformers. The size of the power xformer has much to do with the max output current of the amp's output stage which in turn has an effect on how difficult a load (speaker) that amp can drive.  

Monolith Magnetics makes some very nice (and expensive) transformers. The OPTs I use in my 833C SE amps weigh 62lbs each, with amorphous double-C cores and virgin teflon interwinding insulation. Here's a picture of one during its construction.

http://i1243.photobucket.com/albums/gg559/Quagmire22/833OPTcoilandcore_zps9fb41a95.jpg

They're not cheap but high quality very rarely is.

And yes, the transformers have a huge effect on the sound of tube-based amps, preamps, DACs, whatever.