Grime, 16 14 bytes

Thanks to Zgarb for saving 2 bytes.

e`.^-d\0v#|!..

Try it online!

The e tells Grime to try and match the entire input and print 0 or 1 depending on whether that's possible.

The |! is effectively a "neither" operator, because x|!y is shorthand for (x|y)!. So we make sure that the input neither contains a zero preceded by a non-digit nor is a string of only two characters ([]).

A note about the second half: . matches any character, d matches any digit and - is boolean difference (so we require it to match one character, but disallow that character to be a digit). The only issue is that - has lower precedence than concatenation so we'd have to add parentheses. But Grime has a really nifty feature where you can change the operator precedence. ^- is like - but with higher precedence than any other operator, which is the same as wrapping it and its arguments in parentheses and saves a byte. Likewise the containment operator # has too high precedence and would require parentheses around the stuff in front of it, but we can lower its precedence with v to save another byte.

Grime, 16 14 11 bytes

Thanks to Zgarb for saving 5 bytes.

e`s\0v#|!..

Try it online!

The e tells Grime to try and match the entire input and print 0 or 1 depending on whether that's possible.

The |! is effectively a "neither" operator, because x|!y is shorthand for (x|y)!. So we make sure that the input neither contains a zero preceded by a symbol nor is a string of only two characters ([]).

A note about the second half: P# matches a rectangle that contains at least one match of P. However, in our case P consists of both s and \0 so that would normally require parentheses: (s\0)# (because the precedence of # is too high). But Grime has a really neat feature where you can modify the precedence of operators with ^ and v. So by using v# we lower #'s precedence so that it's lower than that of any other operator (including concatenation), which lets us save a byte on the parentheses.

Grime, 16 bytes

e`..|!.*.^-d\0.*

Try it online!

The e tells Grime to try and match the entire input and print 0 or 1 depending on whether that's possible.

The |! is effectively a "neither" operator, because x|!y is shorthand for (x|y)!. So we make sure that the input is neither a string of only two characters ([]) nor contains a zero preceded by a non-digit.

A note about the second half: . matches any character, d matches any digit and - is boolean difference (so we require it to match one character, but disallow that character to be a digit). The only issue is that - has lower precedence than concatenation so we'd have to add parentheses. But Grime has a really nifty feature where you can change the operator precedence. ^- is like - but with higher precedence than any other operator, which is the same as wrapping it and its arguments in parentheses and saves a byte.

Grime, 16 14 bytes

Thanks to Zgarb for saving 2 bytes.

e`.^-d\0v#|!..

Try it online!

The e tells Grime to try and match the entire input and print 0 or 1 depending on whether that's possible.

The |! is effectively a "neither" operator, because x|!y is shorthand for (x|y)!. So we make sure that the input neither contains a zero preceded by a non-digit nor is a string of only two characters ([]).

A note about the second half: . matches any character, d matches any digit and - is boolean difference (so we require it to match one character, but disallow that character to be a digit). The only issue is that - has lower precedence than concatenation so we'd have to add parentheses. But Grime has a really nifty feature where you can change the operator precedence. ^- is like - but with higher precedence than any other operator, which is the same as wrapping it and its arguments in parentheses and saves a byte. Likewise the containment operator # has too high precedence and would require parentheses around the stuff in front of it, but we can lower its precedence with v to save another byte.