Can A Virus Survive a Factory Reset?

I suggest you use the program created by Etresoft, a frequent contributor.  It will provide a snapshot of your system which we can analyze to possibly determine the cause of your problem. Please use copy and paste as screen shots can be hard to read. On the screen with Options, please open Options and check the bottom 2 boxes before running. Click “Share Report” button in the toolbar, select “Copy to Clipboard” and then paste into a reply. This will show what is running on your computer. No personal information is shown.

https://itunes.apple.com/us/app/etrecheck/id1423715984?mt=12

Use the Add Text button to include the Etrecheck report in your reply.

There are no viruses on a Mac. There are however, a growing number of other forms of malware that won't survive a clean install, but may be brought back into that clean install by a Time Machine Restore which had previously backed them up.

Though we do not recommend any anti-virus, or so-called Mac "cleaner" applications, we do recommend (and use ourselves) MalwareBytes for Mac. Whether you use it in perpetual trial, or optionally, subscribe to the real-time protection it can offer, it just looks for malware and adware. When found, it offers to remove them.

You can limit your exposure to all forms of malware by only downloading software from the actual vendor site, and never from download aggregators (.e.g. Macupdate, CNET, etc.), or Torrent sites. Many websites that do not care about your security, or have a financial reason to host all those bogus download buttons — should simply be shunned.

MalwareBytes should only take about 13 seconds to scan. It sounds like your old machine may need repair or replacement. If you didn’t already have a backup, now is the time to get one.