Should 2 root-canal tooths and a good permanent tooth be fused together like one tooth?

What your dentist has done for you is called "splinting".  This technique of joining multiple crowns together is most frequently done with periodontally involved teeth that are mobile (somewhat loose) because the periodontal disease has destroyed much of the bone support.Root canaled teeth are always weaker than teeth that haven't had root canals.  It appears that your dentist feels that one or more of your teeth are so weak that he wants to protect the tooth and/or root from having to sustain strong loads, possibly causing fracture and loss of the teeth.  That is a judgement call, and without seeing the teeth and x-rays I don't think any dentist could criticize him for doing this for you.  Other options would be extraction, implants, bridges or just doing single (unsplinted) crowns and taking the risk.  I'm sure you would prefer what he did than those other options...right?There are two challenges with splinting multiple teeth together with crowns.  One is that it will be much more difficult for you to clean your teeth with dental floss, thus you may have to be diligent and use a floss threader or superfloss to make sure you remove all of the plaque and don't end up with gingivitis or periodontitis from neglect.  Your dentist should instruct you how to clean and floss around splinted crowns.The other issue is one of "fit".  It is much more challenging for the crowns to fit well at the margins when crowns are splinted together.  If one or more doesn't fit accurately, this could result in leakage and decay that might end up going up under the crown.  Splinted crowns that don't fit properly on one or more teeth can usually be discovered by taking x-rays before (preferably, if the dentist has any doubt) or after permanently cementing the splinted crowns into place.

Splinting the crowns together will work well, but it is guaranteed to fail at some point. Because they are front teeth, the crowns will have to conform tightly with your gums - for this reason you will not be able to clean in between the teeth well, no matter how much you try. Eventually this will catch up with you and you will develop decay at the margin of one of the teeth. When this happens, you need to replace all three teeth. In the long-term it might be less expensive for you to crown #10, and replace #8 and #9 with dental implants.

Connecting teeth together used to be popular in dentistry years ago, for the same reason, they thought it would make teeth stronger. And it kind of did, but there were other problems such as: difficulty cleaning connected teeth eventually leads to cavities and gum disease, during chewing teeth have some natural degree of mobility and when connected rigidly together by crowns/bridges this movement leads to cement breakdown and dental work loosening or cavities under crowns/bridges. So now we try to avoid splinting (connecting) teeth together, but there are cases where it may still be necessary and with proper care and maintanance such teeth can last a long time.