This is a solution to specifically the second part of the question (the largest 5 digit number with the digits 1,3,4,5 (and another of choice) that is a multiple of 12.
Firstly, for a number to be a multiple of 12, it must be a multiple of both 4 and a multiple of 3. For numbers which are multiples of 3, their digits add to a multiple of 3. Thus the 5th number can be either 2, 5 or 8, as this gives a total of 15, 18 or 21 respectively.
Next, I considered the divisibility rule for 4 - The final 2 digits are a multiple of 4 themselves. Thus the last digit is definitely a multiple of 2 at least. A 2 digit number is a multiple of 4 if it’s first digit is even and its second a multiple of 4 (shown by (2k * 10) + 4n = 20k + 4n = 4(5k +n) where k and n are integers) or its first digit is odd and its second a multiple of 4 plus 2 (shown by (2k+1) * 10 +4n+2 = 20k + 4n + 12 = 4(5k + n + 3).
The number picked cannot be 5. If this were the case, the last digit would have to be 4 so that it is even, leaving only odd numbers for the first (of the last two) digit. This as explained above cannot be a multiple of 4 as 4 is not in the form 4k+2 which is required if the preceding digit is odd
This leaves 8 and 2. I initially assumed that 8 would be better as it is larger. With this, it seems the largest possible number is 53184 (84 or 48 must end the number to achieve a multiple of 4 (neither 4 nor 8 are in the form 2k+2 which would allow for the first (of the final two) digit to be replaced with and odd number. 84 is larger than 48 and 531 is the largest of 1,3,5 as it places the highest digits at the greatest place values.
However, I then realised that 2 is a better pick as it is in the form of 2k+2. This allows the first (of the final two) digits to be replaced with a smaller odd number if 2 is placed as the last digit. This leaves larger numbers for the earlier digits. The largest possible number with 1,2,3,4,5 is 54312 and is it uses the smallest odd to pair with 2 for a multiple of 4, and then lists the remaining number (5,4,3) in descending order so that the highest digits are in the greatest place values)
Thus I believe the greatest 5-digit multiple of 12 with these restrictions is 54312.