(Writing without notes here, but I can dig up my sources if necessary.)

There doesn't seem to be any, single well-defined method of crucifixion. At least, so far, no one has unearthed a Roman list of rules and regulations for the procedure.

The original punishment was for the criminal to have a wooden beam put on his shoulders and his hands and arms tied to it. It was not a death sentence, but rather an uncomfortable and public humiliation, rather like being put in the stocks. The phrase pone crucem servo ("Place the cross on the slave") exists from this time as the formal reading of the sentence. Crucifixion has always been viewed as shameful.

(Edit: This was one original punishment. The practice of nailing people up onto boards or trees goes way back. It gets a little confusing because, even in Rome, the term "crucify" also applied to simply tying a person to a stake, or even impaling them on a sharpened stake.)

The punishment could be enforced by either requiring the criminal to go about in public for a certain length of time with the crossbeam tied to him, or the crucified criminal could be hung up on a post in a public place for all to see.

Being hung up in that position (along with being very painful) results in death by suffocation. To avoid this, the feet could be tied to the post to give the criminal some support, or a wooden seat would be attached.

So, hanging a person on a cross does not appear to have been originally designed specifically as a death sentence, but since it could be fatal, it eventually became a torturous method of execution.

When we in the modern world think of a government-ordered execution, we picture an organized and regulated process (like lethal injection) which is carried out in a specific, surgical manner. Crucifixion doesn't seem to have such a specific method.

After the revolt of Spartacus, the Romans crucified thousands of his followers along the road to Capua. One can imagine, in such an assembly-line procedure, that nails might be faster and easier to use than ropes, especially to prevent the victim from working himself loose. Maybe both were used. The basic idea was just to hang the person up there until he died. It didn't matter what you attached him to the cross with.

If you hang a man on a cross by his arms only, he will die in a relatively short amount of time, minutes to hours. If you support his body, give him a seat to perch on or attach his feet to the upright, he will live longer, sometimes for days. This was no doubt taken into account by those carying out the sentence. Adjusting the position of the victim's feet or giving him a platform to sit on could result in a quicker or slower death. Nails are certainly more painful than ropes and, if the crucifixion is to be a death sentence, you might as well use nails.
There was also the procedure known as the crurifragium, breaking the legs of a man nailed to a cross, leaving him hanging by only his arms and suffocating him. This is what the Romans would do if the man was still alive after hanging there for a time and the Romans wanted to get it over with. A process which has never ceased to give me the willies when I think about it.

About the cross itself. It would almost certainly be in the shape of a capital T. This is because it is the simplest thing to construct. A vertical pole is sunk into the ground, the crossbeam has a mortise carved into it to fit on top of the pole. You tie, or nail, the condemned to the crossbeam and lift him up onto the pole. The height of the thing wouldn't have been more than six or seven feet. Any taller, and it would be too difficult to get the crossbeam up there without a ladder. (Jim Bishop mentions a crux sublimius, a very tall pole used for important public executions, but I haven't followed up on that myself.)

The most famous victim is, of course, Jesus, and there is a wealth of artwork devoted to his crucifixion. The crosses in those paintings are rarely historically accurate. Constructing a cross as depicted in a Renaissance painting would be a needlessly complicated work of carpentry. The images of Jesus carrying a large, complete, cross along the Via Dolorosa are almost certainly incorrect. Again, it's just too complicated a device. The term "cross" (crux) was always a reference to the crossbeam. It's only in our modern image of "Christ on the Cross" that we think of the entire structure as a "cross."

TL;DR "Ropes or nails?" We don't know because there wasn't a specific method of crucifixion. In the case of Jesus, it was probably nails.

I'd also be interested to hear someone knowledgeable about the subject weigh in on the possible/probable shape of the cross.

Important to note that the practice was not Roman, but from the middle East, and they imported it after conquering the region. It was also not something that would be done to Roman citizens with rare exceptions.

It had a religious basis if memory serves, to prevent the criminal from tainting the sacred earth. Considering how old the practice is ropes were more likely than not used, also considering how expensive metal was, doubt they would bother wasting nails on criminals aside from trying time to time to make an example out of someone, if at all.

Mentions about nailing were mostly from judeo source Josephus.