Does the fact that I install unnecessary modules make PHP overall slower? e.g. php-tidy, php-xmlrpc, php-bcmath.
I use the latter only on very rare occasions. Would like to have it (and at least 5 other extensions) only if that doesn't mean any significant system resource usage increase.
We are talking about a server that's handling 200 000 to 500 000 requests daily.
Run slower? Likely not. Take a larger amount of memory? Yes. If your system is memory constrained, and that's leading to performance issues, then reducing the number of modules may help, but otherwise, I don't think it's worth doing anything.