Learn English – what is the meaning of the phrase “nice shot”

Beat the band is an idiom for to the greatest possible degree.

It's like they're smiling as big as can be.

It's another form of "to beat all", akin to "that beats everything".

From the American Heritage Dictionary in Dictionary.com:

Idioms & Phrases
to beat the band

Also, to beat all. To the greatest possible degree. For example, The baby was crying to beat the band, or *The wind is blowing to beat the band , or *John is dressed up to beat all . This idiom uses beat in the sense of "surpass." The first term may, according to one theory, allude to a desire to arrive before the musicians who led a parade, so as to see the entire event. Another theory holds that it means "make more noise than (and thereby beat) a loud band." [Colloquial; late 1800s]

It has another meaning as well, which is not used in your example:

to beat the band
very briskly; very fast. He's selling computers to beat the band since he started advertising. She worked to beat the band to get ready for this.

Shot in ODO

1 (of coloured cloth) woven with a warp and weft of different colours, giving a contrasting effect when looked at from different angles:
a dress of shot silk
- interspersed with a different colour: dark hair shot with silver
- (shot through with) suffused with (a particular feature or quality): the mist was shot through with orange spokes of light

It's not a repeated pattern; it's a highlight of some sort, either the odd thread, or the entire warp or weft.

In the quote, it could be either, but since a different warp and weft would give rise to "a truly beautiful shade," it's probably that: woven throughout with two different colours.

I couldn't find an example of grey and red, but here's blue and red from the blog Snoring Scholar: