Hi skichick97701,
Thanks for coming to Apple Support Communities for help today. It sounds like you are concerned about updating your software on your Mac.
You shouldn't need to upgrade to Catalina in order to sync your iPhone. Your computer will easily support the macOS upgrade though. You can read more about Catalina here: macOS Catalina. If staying on Mojave, it would be advisable to check for an iTunes update available in your Mac App Store.
What you may be seeing is this message: If you see "Software update is required to connect to your iOS device" on your iPhone, iPad, or iPod touch. This would just be a small update to make and should have no effect on the performance of your Mac.
You can also read more about your battery here: About Mac notebook batteries. If you Mac needs battery service, there should be a message displayed.
You can also see how pages load when the Mac is in safe mode: Use safe mode to isolate issues with your Mac. If pages are loading faster in safe mode, it might be due to an extension that Safari is using: Get extensions to customize Safari on Mac. Extensions and other third-party programs are disabled in safe mode.
Have a fantastic weekend!