Perhaps an attorney could explain it better, but my understanding of the homestead exemption is that it exempts the equity up to the limit. So, if your equity in your home is $100,000 and you sell it, you can exempt the first $60,000, but the other $40,000 potentially is fair game. The collection agency would have to get a judgment and the slap a lien against your house.
Please note that the existence of such a lien might "scare off" many potential buyers, should you decide to sell your home.
Now, if the equity that you have in your home is less than $60,000, you probably have nothing to worry about.
I have not researched the statute of limitations (SOL) for TX. But, virtually everywhere the SOL is an affirmative defense that you have to raise in court IF you get sued. Therefore, the collection agency can, in fact, sue and seek to get a judgment against you. If you ignore the summons and do not raise this defense, the collection agency could indeed get a default judgment against you. The SOL does NOT protect you from getting sued. It only gives you a defense that you must assert IF you do get sued. You would also have to present arguments as to why you think the SOL starts on a more remote date in the past than the collection agency thinks/states.
That being said, going to court is expensive for a collection agency especially when there is no real certainty of a quick "payoff". Collectors say lots of things to get debtors to send the rent or mortgage money toward the debt which they are trying to collect on. Most of these people work on commission. They don't care one "twit" about your other obligations. Sometimes these collectors "stretch" the truth.
If you don't find that this collection agency is willing to work with and talking to them over the phone is simply "raising your blood pressure", then send them a certified letter, return receipt requested, stating that they are only to contact you in writing about this debt.
Be vigilant. Do NOT ignore any court summons that you should recieve. But, I would guess that most of these threats are "bluffs" and that you won't get sued. Obviously, I cannot guarentee that the collection agency won't make good on its threats.