The short answer is No. search.log files are not stored under $SPLUNKHOME/var/log/splunk/ but are written to SPLUNKHOME/var/run/splunk/dispatch// .
Scheduled jobs (scheduled saved searches) include the saved search name as part of the directory name.
Search jobs manifest as a process in the OS. There are two processes in Linux for each search job: search-launcher and process-runner. You can isolate all the Splunk search processes with: ps -ef | grep search. The main job is the one using system resources and contains search --id in its name.