In most cases yes SMS messages can easily be traced if they initiated from an actual phone and not an International bulk SMS relay service (in many cases those are also still traceable).
Many countries require that all telephone metadata (which includes the source and destination numbers) is recorded and kept on file by the telephone provider either for a set time length or in some cases indefinitely. Likewise for billing purposes most telecoms keep track of every individual connection for what each one of them deems to be a reasonable time period even if it's not mandated by law in their jurisdiction.
The longevity of these records is also rapidly increasing as the price of storage declines and companies recognize the value of data mining, leveraging Big Data in numerous ways, and having large amounts of "learning data" for Artificial Intelligence applications.
Gathering the data to do what you have mentioned would not require hacking your phone but it also doesn't mean that did not happen. Most likely this information is available publicly or the bully had access to that information via some other means.
Depending on where you live law enforcement may be able to help. Likewise in most areas you can hire a Private Investigator locally who may be able to help. If you are still not sure where to go most attorneys give a free initial session and may be able to refer you to a specialist in your area.
Attorneys can also help you draft a discovery request which in some cases may help with getting research started, or at least preserved, relating to your incident.
Additionally contact the phone company and ask them to preserve the records and see if they can help. They may not be able to divulge more information without a court order however in some cases their internal fraud/abuse teams may be able to rapidly escalate getting the information to local law enforcement.
Note: As much as I wish this were not the case keep in mind that most law enforcement offices are overwhelmed by on-line crime and may consider this lower priority when compared to dealing with other issues that are either deemed more important or easier to solve. In their defense, there is a LOT of on-line crime and they have very little resources (and in some cases little training ) to deal with it. Be as helpful to them as you can be. Finally, having an attorney involved may help a lot in this regard if nothing else it may prioritize your case over others where no attorney is present.
I would also recommend pursuing the e-mail as well. In some cases this will make it much easier to identify the person doing this.