You can use the SUBSTITUTE function to replace the *s with blanks. Excel will then treat the result as a number if you use the addition (+) operator as in your examples. This will not, however, work with a function like SUM, because SUM treats any string as zero. I.e. 6+SUBSTITUTE("5*","*","")+4+SUBSTITUTE("7*","*","") does equal 22, but SUM(6,SUBSTITUTE("5*","*",""),4,SUBSTITUTE("7*","*","")) equals 10.
If you wish to use the SUM function on a range of values including the strings with trailing * chanacters, you can use an array fuction:
=SUM((SUBSTITUTE(A1:A4,"*","")*1)
After typing or copying the formula, press Ctrl-Shift-Enter together rather than just Enter. This will cause the formula to display in the formula bar surrounded by curly braces, signifing that it is an array function, i.e.:
{=SUM((SUBSTITUTE(A1:A4,"*","")*1)}
This works by using SUBSTITUTE on each value and then multiplying that value by 1, which causes the product to be a number before it is SUMmed.