(sed  -e'y/)},={/(((((/'
  -e's/-([^(I]*)[^0-9]*([0-9]*)[( ]*/- 2 -1/;=' |
paste -d- - - |
sort  -t- -nk3,3 -nk1,1 |
sed   -e's/^[^-]*-//;:n' -e'h;$!N'
  -e's/(-([^-]*-).*[^ ]) *n([^-]*-){2}2/1 - 3/;tn'
  -ex;:t -e's/(([^-]*-)[^/]*) - *2/1,/;tt'   -e'p;g;D' )
< in.txt > out.txt

6/13/2015 12:00:47 AM - {   562} START Web
6/13/2015 12:00:47 AM - Requested Web connection from 123.125.71.103[123.125.71.103], ID=562
6/13/2015 12:01:24 AM - {   563} START POP3
6/13/2015 12:01:24 AM - Requested POP3 connection from 10.127.251.37 [10.127.251.37], ID=563
6/13/2015 12:01:24 AM - (   563) USER test.mail@test.me
6/13/2015 12:01:24 AM - POP3 connection with 10.127.251.37 [10.127.251.37] ended. ID=563
6/13/2015 12:01:24 AM - {   563} END POP3
6/13/2015 12:01:24 AM - {   564} START POP3
6/13/2015 12:01:24 AM - Requested POP3 connection from 10.127.251.37 [10.127.251.37], ID=564
6/13/2015 12:01:24 AM - (   564) USER test.mail@test.me
6/13/2015 12:01:24 AM - POP3 connection with 10.127.251.37 [10.127.251.37] ended. ID=564
6/13/2015 12:01:24 AM - {   564} END POP3
6/13/2015 12:01:40 AM - Web connection with 123.125.71.103 [123.125.71.103] ended. ID=562
6/13/2015 12:01:40 AM - {   562} END Web

6/13/2015 12:00:47 AM - 562 - START Web, Requested Web connection from 123.125.71.103 [123.125.71.103] - 6/13/2015 12:01:40 AM - Web connection with 123.125.71.103 [123.125.71.103] ended., END Web
6/13/2015 12:01:24 AM - 563 - START POP3, Requested POP3 connection from 10.127.251.37 [10.127.251.37], USER test.mail@test.me, POP3 connection with 10.127.251.37 [10.127.251.37] ended., END POP3
6/13/2015 12:01:24 AM - 564 - START POP3, Requested POP3 connection from 10.127.251.37 [10.127.251.37], USER test.mail@test.me, POP3 connection with 10.127.251.37 [10.127.251.37] ended., END POP3

6/13/2015 2:15:09 AM - Starting delivery thread, ID=667
6/13/2015 2:15:09 AM - {   667} START Delivery
6/13/2015 2:15:09 AM - Active delivery threads (ID=667 start): 0
6/13/2015 2:15:09 AM - [   667] Delivering to test.mail@test.com
6/13/2015 2:15:09 AM - [   667] Forwarding mail...
6/13/2015 2:15:09 AM - [   667] Forwarding message to 1 address(es)
6/13/2015 2:15:09 AM - Delivery thread, ID=667, terminated after running for 0.031 seconds.
6/13/2015 2:15:09 AM - {   667} END Delivery
6/13/2015 2:15:09 AM - Active delivery threads (ID=667 end): 0

6/13/2015 2:43:24 AM - Exended logging
6/13/2015 2:43:24 AM - Setting msgSizeLimit 15

(   sed -e= -e's/- [({[] *([0-9]*) *[])}]/- 1 -/;t'
            -e's/-(.*[^, ])[, ]*ID=([0-9]*)[, ]{0,1}/- 2 -1/' |
    paste   -d- - - |
    sort    -t- -nk3,3 -nk1,1 |
    sed     -e's/^[^-]*-//;:n' -e'h;$!N'
            -e's/(-([^-]*-).*[^ ]) *n([^-]*-){2}2/1 - 3/;tn'
            -ex;:t -e's/(([^-]*-)[^/]*)- *2/1:::/;tt'   -e'p;g;D'
) <in >out

6/13/2015 12:00:47 AM - 562 - START Web ::: Requested Web connection from 123.125.71.103[123.125.71.103] - 6/13/2015 12:01:40 AM - Web connection with 123.125.71.103 [123.125.71.103] ended. ::: END Web
6/13/2015 12:01:24 AM - 563 - START POP3 ::: Requested POP3 connection from 10.127.251.37 [10.127.251.37] ::: USER test.mail@test.me ::: POP3 connection with 10.127.251.37 [10.127.251.37] ended. ::: END POP3
6/13/2015 12:01:24 AM - 564 - START POP3 ::: Requested POP3 connection from 10.127.251.37 [10.127.251.37] ::: USER test.mail@test.me ::: POP3 connection with 10.127.251.37 [10.127.251.37] ended. ::: END POP3
6/13/2015 2:15:09 AM - 667 - Starting delivery thread ::: START Delivery ::: Active delivery threads (start): 0 ::: Delivering to test.mail@test.com ::: Forwarding mail... ::: Forwarding message to 1 address(es) ::: Delivery thread terminated after running for 0.031 seconds. ::: END Delivery ::: Active delivery threads (end): 0