There are 9*10^4 five digit numbers with no leading zero.
There are 7*8^4 five digit numbers with no leading zero, containing neither 3 nor 7.
So there are 90000-28672 =61328 numbers satisfying condition 1.
Roughly 1/37 of these numbers i.e 1657 are divisible by 37.
I am happy with this number as an engineer's approximation.
When some solver evaluates the exact number we will comment
upon the accuracy of my approximation.