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Here is a quick rundown of the Texas laws on carrying, as I understand them. I am not a lawyer, and I strongly suggest you read Chapter 46 of the Penal Code, along with sections 30.06 and 30.07. Then read the parts of chapter 411 of the Government Code that deal with the license to carry. It is always best for you to read the law yourself. Second best is to pay for professional legal advice.

These are all very broad generalizations and oversimplifications but will give you a general idea of the law. The law is, in many case, very finely nuanced and not as general as I will make it sound.

In Texas, it is generally illegal for a person to carry a gun without a license if he is off his property and out of his car. Anyone can carry a gun anyway they want on their own property. It is legal for a person to carry a gun in their car if it is concealed, and that also allows them to carry to or from their car. Anyone with an LTC (license to carry) may carry a pistol almost any way they want as long as it is concealed. A person with an LTC may carry a pistol unconcealed if it is in a belt or shoulder holster. A person with an LTC may not carry a pistol where there are proper signs forbidding it, with a separate sign required forbidding either concealed or open carry (two signs to forbid both). As you noted though, you cannot post those signs).

There are almost no restrictions on a person carrying a long gun (rifle or shotgun).

People with an LTC are required to show it to a peace officer or a magistrate under certain conditions. Unless you meet both the position and the other requirements, no one has to tell you or show you that they have an LTC.

A general conclusion would be that it is legal for anyone to carry a rifle into your library and for anyone with an LTC to carry a pistol, either openly or concealed, into the library. You can ask if they have an LTC but they are not required to answer you.

But now for some good news, and this might help you feel a little more secure. First, I want to point out that mass shootings do not occur very often. They are, in fact, a very rare occurrence. They are so rare that it takes several years before criminologists can get enough incidents to get a valid scientific study. Contrary to the mass media's best efforts to portray otherwise, the odds are overwhelmingly against you ever seeing or being involved in a shooting like this.

Secondly, again contrary to mass media impressions, people with LTCs are among the best people in the world. They tend to be nicer and more polite (well, my experience says so) and they have much less criminal activity. There are fewer people with LTCs convicted of crimes each year than there are police officers. Part of this is the personality of the people who get licenses. They are law abiding citizens. If they weren't they would not bother to get the license and would just carry the gun anyway. Part of it is that they have some very tough standards to pass to get the license, including a background check with the FBI running their prints.

And finally, with a few exceptions, people are very unlikely to be carrying rifles or shotguns around. They are just not too convenient. They are bulky and interfere with your normal motions. The exceptions include people who are carrying the weapon to protest something - mainly weapons laws but we have seen a few protesting other things recently.

The final question you mentioned was about what to do if you see someone carrying a long gun coming your way. My first gut instinct is to say that if he has it on a sling, you probably have little to worry about. If he is carrying it in his hands, you may have something to worry about. If you see him coming in time and can lock the front door before he gets there, I could see that as a reasonable action. You do not have to let him in the building. You should also call the police and let them know what is happening. The reason I mention carrying in his hands or on a sling is that Disorderly Conduct, section 42.01 of the Penal Code, makes it a crime to carry a weapon "in a manner calculated to alarm." If he is scaring you with it, it is time to let the police know and handle it. They may not be able to do much (their or the DA's definition of alarm may differ from yours) but they can probably convince the person to leave.

Hope that helps a little bit. I am sure a few others will chime in to clarify points I got wrong or glossed over too much.