No idea if you have grounds to successfully sue for anything since you posted no facts about the issue. Repost with some case facts if you want a meaningful answer.
NOT LEGAL ADVICE. FOR EDUCATION AND INFORMATION ONLY. Mr. Rafter is licensed to practice in the Commonwealth of Virginia and the US Federal Courts in Virginia. There is no implied or actual attorney-client relationship arising from this education exchange. You should speak with an attorney licensed in your state, to whom you have provided all the facts before you take steps that may impact your legal rights. Mr. Rafter is under no obligation to answer subsequent emails or phone calls related to this matter.
Yes, you can sue. The Texas constitution guarantees all citizens free access to our courts. This means that anyone can sue anyone, at any time, anywhere, for any reason, for any amount of money and without first proving anything to anyone. However, filing a lawsuit and winning one are two entirely different things. Winning a lawsuit and actually recovering some money are also two entirely different things.
I cannot comment on which court system is most appropriate for your case without knowing what the facts are.
I strongly suggest that you consult a lawyer. A lot of folks sue pro se in small claims court, but only a tiny percentage of them ever end up actually recovering any money. Most of them lose because of a simple lack of knowledge of how to prosecute and try a lawsuit. And those few who win get only a piece of paper called a "judgment." Converting the judgment into money requires knowledge of the law and procedures regarding post-judgment collection efforts.
Considerably more information is required. What is the basis of your claim for breach of contract? What did the other promise that they failed to do and how did it cause you to suffer damages? What are the facts related to "false advertising".
It is not likely you will be able to recover mental anguish for a breach of contract, Texas law does not allow that recovery.
If you have a claim for a tort, such as negligence, you can recover mental anguish caused by the tort provided you can prove the defendant's negligence proximately caused the mental anguish and you can prove the mental anguish you suffered by something more that I was upset. You don't suggest any facts that would support any recovery for mental anguish in your post.
A claim in small claims court will also have limited damages, usually $5000 or less. There is no way to tell you whether you have any viable claims or not, where you should sue if you have viable claims and whether you have any real chance of prevailing. You need more facts and you need to call a lawyer and discuss those facts with him.
This answer does not constitute legal advice or create an attorney-client relationship.
Based on what you wrote:
No, you cannot sue in Federal Court because there is no federal question or diversity jurisdiction.
Yes, a small claims court can award money for mental anguish. However, you probably will not get any money for mental anguish just because someone breached a contract or falsely advertised.
You probably cannot sue for false advertising.
This is from a Texas-licensed lawyer practicing from near St. Louis in the "show me" state. I think even in Texas, and even in small claims court, the judge is going to say "show-me" the evidence and "where's the beef?" If you come into a Texas small claims court with a mental anguish count in a breach of contract case, the judge will laugh and dismiss that count - if you are lucky.
Keep in mind that although my colleague very correctly states "... anyone can sue anyone, at any time, anywhere, for any reason, for any amount of money and without first proving anything to anyone." the rules of practice allow sanctions for frivolous lawsuits, which can include paying all of the legal fees of the person(s) you frivolously sue. Judges are not fond of people who come in with ridiculous claims and waste the court's time. Suing for mental anguish over a false ad or breach of contract risks being found frivolous. Don't do it. See a lawyer first to see if you have some sort of tort you can use as a means to non-frivolously claim mental anguish.
I am not your lawyer and you are not my client. Free advice here is without recourse and any reliance thereupon is at your sole risk. This is done without compensation as a free public service. I am licensed in IL, MO, TX and I am a Reg. Pat. Atty. so advice in any other jurisdiction is strictly general advice and should be confirmed with an attorney licensed in that jurisdiction.