iMac 2017 keeps crashing

Thanks for the report.

Failing hard drive - This computer has a hard drive that appears to be failing.

The reason your iMac is slow is due to the 5400-rpm hard drive, and it also appears to be failing! Luckily, you can use an external SSD as your startup disk to run macOS and all your data from that. It will make your Mac 10 to 30 times faster for storage, and in turn, make your entire Mac system much faster. For more info, instructions, and what external SSD to buy, please see: How to Setup and Use an External SSD as your Startup Disk on a Desktop Mac - Apple Community.

Another option is to replace this Mac with a new Apple Silicon Mac: Mac - Apple. If you go this route, it is generally recommended that you purchase 16GB of Memory with the largest SSD that you can afford. You can also use Apple Trade-In to get credit towards a new Mac: Apple Trade In - Apple.

No Time Machine backup - Time Machine backup not found.

However, since your drive is failing, you need a backup! To start, you need an external hard drive. Here’s a few excellent ones sold by Apple: 

• LaCie Mobile Drive 4TB External Hard Drive USB-C - Apple
• LaCie 4TB Rugged USB-C Portable Hard Drive - Apple

You can then follow the steps in this Apple Support article to setup your Time Machine backup: Back up your Mac with Time Machine - Apple Support. 

Also, moving forward, I recommend that you use iCloud in addition to Time Machine to sync all your files and photos to all your Apple devices. Although iCloud is not a "true" backup service, iCloud can also act as an emergency off-site backup, but this is why iCloud should be used in addition to Time Machine. Here are some resources on setting up iCloud for Mac:

• Upgrade to iCloud+ - Apple Support
• Set up iCloud Drive - Apple Support
• Add your Desktop and Documents files to iCloud Drive - Apple Support
• Set up and use iCloud Photos - Apple Support

After you setup iCloud, turn off "Optimize Mac Storage" in System Preferences > Apple ID/iCloud. That way, all changes made to files in iCloud are also backed up to Time Machine. If you have this setting on due to a limited amount of storage on your Mac, that is fine. Just know changes won't be backed up to Time Machine, only iCloud.

Cheers,

Jack

Welcome to the Apple Support Communities.

I believe we need to proceed with an EtreCheck report. EtreCheck is a useful App to diagnose and identify issues, both hardware and software. It is trusted by many users. Please follow the steps outlined below:

1. Navigate to: https://etrecheck.com and get the free version of EtreCheckPro.
2. Download it. It is a free and trusted app. It is often recommended here on Apple Support Communities. 
3. Open EtreCheck, and before running the report, be sure to check “Allow full Drive Access” at the bottom of the EtreCheckPro window.
4. Run the report, and put it in a post, as shown here: https://discussions.apple.com/docs/DOC-250000211. We will then examine the report for issues in your Mac.

Cheers,

Jack