I can only support you on workdays, on the weekends my family needs me, sorry about that.
1. You can disable search in posts and pages, which is enabled by default, and select the songs custom post type to search.
2. Slow ajax response in most cases is not related to the search, there are many factors influencing the search speed.
Please read the Performance tuning section and the tips & tricks section on how to increase your site performance.
Moreover I did run some tests on your site, and there is something definitely clogging the ajax responses (not only ajax search, but every ajax operation).
Look at this test. The performance tracker shows that the search is finished after 0.2 seconds in reality. However most likely something else (another plugin or theme etc..) is keeping your server busy, because the results appear only after 14!! seconds, and that's the same exact query on both screenshots. That means that something is most likely making 70 times more operations before or after the search is finished.
Try to disable all the plugin and switching the theme and check if it improves the performance, it definitely should. If not, then I might need ftp access as well to check if there is any connection for the slowdown with the search plugin.
Interestingly, repetitive search shows various performance. I've got results for the search term “test a” in 14 seconds, then in 2.5 seconds for the second time, then 5.5 seconds for the third time, then 2.5 seconds for the fourth time. Is it possible that your hosting provider might throttle your server performance? I don't see any other explanation on why the same exact term delays 14 seconds, then only 2.5 seconds.Best,
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