I’d like to help you, but the troubleshooting steps I’ve outlined require FTP and phpMyAdmin access, which means if I did something stupid I could completely destroy your site. I don’t want that responsibility. You need to be very careful who you give such access to.
Since your site is fairly new, instead of spending a lot of time trying to figure out what went wrong, it may be easier to handle this as a hack (I still do not think this is a hack) and wipe everything, do a complete reinstall, then restore the DB from a backup made before the trouble started. If you do not have such a backup, before wiping everything, export only the posts table and only images in the uploads folder that you know are part of the posts. Everything else will be lost.
By limiting what content is used from the possibly hacked site, you minimize the chance of reintroducing malicious code when the data is imported into the new installation. The risk of reinfection is still there though. To be completely safe, consider starting over with new content and use nothing from the possibly infected site.
To be safe just in case there was a hack, also change all passwords before doing anything else, then change them again when the new installation is complete. More information on dealing with hacks is available from FAQ My site was hacked.