Ok, the first thing I would do is check out this:
http://wiki.gbatemp.net/wiki/List_of_3DS_homebrew
It's a list of all available homebrew/HBL exploits/CFW - find an entry point that will work for you. that's step 1 (click the thread link to the right of the homebrew in the list to find the download for them)
I believe you can use browser hax on 9.5 but am not 100% sure. I also recommend getting the homebrew starter pack and dragging everything to the root of your 3ds SD:
http://smealum.github.io/3ds/
If you find an entrypoint that doesn't use a game save like oot3dhax or smashhax I would install one on a cartridge (you can back up your save using svdt homebrew or powersave)
Yes OOT works with 10.6 but you'll need Powersaves to get it to work if you don't have an alternate method of getting to the hb menu.
Yes. I've updated with non-legit cias and nothing happens. But yeah. Downgrade first.
I agree with this the first thing I would do personally is downgrade to 9.2 using the appropriate sysupdater or safesysupdater Link:
https://gbatemp.net/threads/wip-safesysupdater.409392/ then setting up an emuNAND with CFW that you can update to 10.7
You will also need to find the appropriate 9.2 files for your 3ds to downgrade. If they are not good safesysupdater will catch them and we can not post links here but google is your friend. it'll be named something like Old3ds_9.2 (FULL)
Once you have successfully downgraded (you may need to force shut down and remove SD card if it does not reboot automatically or won't boot up after downgrade is successful shows up)
then I would look into a good CFW to use and install it. there are tons of
tutorials showing you how to do so. I used this one on my sister's 3ds and it's working very well:
if you have any questions then feel free to reply to my post and I'll try to help you out.
Good luck and have fun.