selecting operating system to boot

First of all the performance issue is most likely being caused by a failing hard drive or an issue with third party software. I would first try running EtreCheck and posting the report here so we can examine the report to see if there may be any software issues. Plus the report may include summaries of some system logs which may provide clues. Post the report here using the "Additional Text" icon which looks like a piece of paper.

After running EtreCheck, then you can try running the Apple Diagnostics to see if any hardware issues are detected. Unfortunately the diagnostics don't detect most drive failures. You can run DriveDx to check the health of the drive(s). Post the complete text report here using the "Additional Text" icon which looks like a piece of paper.

As for your failed installation of El Capitan, it is because you didn't really run the macOS installer. What you did was download the macOS El Capitan installer as a .dmg file which contains an app (.pkg file) which is used to extract & create the real El Capitan installer to place it into the Applications folder.

In order to downgrade macOS from Mojave to El Capitan will require you to create a bootable macOS 10.11 El Capitan USB installer using the instructions in this Apple article:

How to create a bootable installer for macOS - Apple Support

This article also explains how to extract the macOS 10.11 El Capitan installer from the downloaded .dmg file. After booting from the macOS 10.11 USB installer you will need to use Disk Utility to erase the whole physical drive as GUID partition and MacOS Extended (Journaled). Then quit Disk Utility, then select the "Reinstall macOS" option. Erasing the drive will destroy all data on the drive, so make sure to have a good backup of your data before doing this.

You should always have frequent and regular backups of your computer and all external media (including the cloud) which contains important & unique data.

Have you considered using an external SSD for improved performance? While the hard disk found in your Mac is perfectly suited to light tasks, such as email and web browsing, but for more advanced tasks, and for users who want the maximum performance from their Mac, this hard drive does have its limitations. An external SSD will make your Mac 10 to 40 times faster for storage, and in turn, make your entire Mac much faster and even better. To learn more, see: Use an external SSD as your startup disk … - Apple Community. Additionally, you could upgrade to macOS Catalina, the latest version of macOS supported by your Mac, and have excellent performance if you setup an external SSD.

However, if you're set on going back to OS X El Capitan, create an OS X El Capitan bootable installer by following these steps:

How to create a bootable installer for macOS - Apple Support. If you need a storage device for a bootable installer, here's an option: SanDisk 64 GB iXpand Flash Drive Go - Apple.

You'll also need a backup, since doing this will erase all data. However, you cannot use Time Machine because Time Machine isn't compatible with previous versions of macOS than what version is that's used for the backup. So, I'd suggest you use an external storage device (formatted as Mac OS Extended Journaled since APFS isn't recognized in OS X El Capitan), and manually make copies of your data you want to save:

• Drag and drop items on Mac - Apple Support
• Use external storage devices with Mac - Apple Support

If you need another external storage device, this specific external hard disk, sold by Apple, is an excellent option:

SanDisk Professional 1TB G-Drive ArmorATD™ - Apple.

Once you have a bootable installer and copies of your data, you can erase your disk and reinstall OS X El Capitan. Shut down your Mac, turn it on, and immediately press and hold the "option" key on your keyboard. Select the "Install OS X El Capitan" bootable disk.

Next, Use Disk Utility to erase an Intel-based Mac - Apple Support. When erasing your Mac, select the main Apple internal drive. Its name will end in "Media". Make sure this information matches:

Name: Macintosh HD

Format: OS X Extended (Journaled)

Scheme: GUID

After erasing your startup disk, Quit Disk Utility, and select “Reinstall OS X”.

Setup your Mac again, and copy and paste (How to copy and paste on Mac - Apple Support) your data from the external disk back to your Mac.

Jack

Hello!

What model of iMac do you have?

• Identify your iMac model - Apple Support
• Find the model and serial number of your Mac - Apple Support

Jack

If you are on Mountain Lion, then you may need to partition & format the USB stick instead of just erasing it. Within Disk Utility click on the physical USB stick and select the "Partition" tab. Select "1 Partition" from the drop down list box. Make sure the file system is MacOS Extended (Journaled). Click the "Options" button and make sure "GUID Partition" is selected. Verify that MacOS Extended (Journaled) is still selected as the file system if you had to change to GUID partition. Give the volume a name which does not contain any spaces or special characters as it may make using the command line more complicated for you when creating the bootable macOS 10.11 USB installer. Here is an article which shows the process and provides pictures:

https://eshop.macsales.com/tech_center/formatting/Mac_Formatting_6-10_R3.pdf

FYI, the process is a bit difference when booted into macOS 10.11+.

Also, make sure to read the instructions very carefully for creating the bootable macOS USB installer since you must first extract the real installer from the downloaded .dmg file and included .pkg file/app.

I'm glad you were at least able to get some form of macOS reinstalled.