You can do multiple -replace statements within a single command, so for instance if you just wanted the client name from the middle you'd use :
$string="ABC3007E 02072015 01:36:49 <11.10.14.10 /492C> Zumb Client --! STOPPING CURRENT PATH OF OBJECT RESOLUTION"
$short = $string -replace "--!.*" -replace ".*> "
$short
However in those cases it's nice and simple since you have definitive characters/strings to find and remove everything before or after. Just getting "ABC3007E, 11.10.14.10 and the string after --! " would be harder, since the other text will obviously
change so you'd need to look at replacing the number of characters, and some of those strings are presumably variable length.
You'd probably need to experiment with .SubString to get the bits you wanted, and potentially pull out the required elements and then put them back together in the format wanted.
May be worth checking out this post http://www.lazywinadmin.com/2013/10/powershell-get-substring-out-of-string.html which gives some good examples of the kind
of things you can do with strings, and the multiple ways you can approach it depending on personal taste.