If you MUST keep them then I would suggest getting an IT professional to check the PC over.
Reformatting by the way will not achieve much that a thorough clear-out of temporary files and redundant programmes won't also do in much less time. Windows own disk clean-up can do a lot and free programmes like ccleaner can do more. Back everything up first though (see later)
For a non SSD laptop being able to actually use the PC within 2 minutes of 'cold' power-on is quite good and many take 4 - 5. To be usable from cold in 1 minute or less you need a SSD drive, fast motherboard, memory and processor and windows 10.
You may also find your windows version is a makers (OEM) edition and if you do 'reformat' the entire drive it is likely to have many driver and licence issues. The better process is to save everything elsewhere and then use the laptop "factory restore" process.
Regarding Yahoo mail if you only access it as webmail (using internet explorer you say) the messages are probably all up there in the yahoo servers. Yahoo mail folders people mention are usually sub-folders for the server inbox etc and are also on yahoos servers.
Use a trusted other persons PC to log on as you normally do on yours (don't save password) and check exactly what is on line
If you use or have previously used an off line mail client like Window mail, windows live mail, outlook, outlook express etc it is likely some or all of the emails have been downloaded (and maybe deleted from yahoo servers) and are stored on your hard disk. The folders may be called Yahoo mail or anything else.
Folders on the PC (and associated mail contact lists) can be easily backed up elsewhere too and then substituted for the default files of the same programme on a re-built PC. You can also use a mail client like outlook to download (copy not delete!) everything that is on Yahoos servers and store these in an accessible 'offline' format on a PC
Knowing what you are doing and if it 'looks right' is key or it won't work. As you say the messages are critical I'm not going to say more.
Get a pro to find any critical files on your Laptop and back them up for you and demonstrate how to access / restore them. If the laptop stops working you may very glad you did/have this.
The same professional can quickly create a backup of the entire laptop harddrive that would mean if whatever you do next goes wrong you can restore everything as it was before. . One way of doing this economically is to remove the existing drive and make it your backup in a drawer. Then 'clone' it onto a new drive (can be a SSD or mechanical drive.
People like to swap to SSD's but my experience is that if the existing drive is large and full and/or the PC (especially the processor, memory and motherboard) is actually not that fast you will pay a lot for a SSD and the Laptop overall will maybe start 1 minute faster. This can be a bit disappointing if you bought a 1TB laptop SSD
Again whoever is going to do the backup /swap can inform you of the likely outcome
Last edited by Jetstream67; 6th May 2019 at 11:44.