That Christ's hands were nailed to the cross is firmly established by Scripture. Following the Resurrection, the Apostle Thomas doubted that Christ was alive:
So he [Thomas] said to them, "Unless I see in His hands the print of the nails, and put my finger into the print of the nails, and put my hand into His side, I will not believe." (John 20:25, NKJV)
When Christ appeared to Thomas, he instructed
Reach your finger here, and look at My hands; and reach your hand here, and put it into My side. Do not be unbelieving, but believing. (John 20:27, NKJV)
While this does not rule out that His arms were also bound, it does require that they were nailed.
The Romans used a variety of different shaped crosses to execute criminals. For example, Seneca the Younger, a contemporary of Christ, wrote
I see before me crosses not all alike, but differently made by different peoples: some hang a man head downwards, some force a stick upwards through his groin, some stretch out his arms on a forked gibbet. (Moral Essays 6.20)
Josephus, who lived just after Christ, records
the soldiers, out of the wrath and hatred they bore the Jews, nailed those they caught, one after one way, and another after another, to the crosses (The Wars of the Jews 5.11.1)
So it seems that they had a variety of choice and applied them as they saw fit. The typical crosses were |-shaped, T-shaped, +-shaped, and X-shaped. Scripture does not clearly state which shape was used, but Christian writings in the first few centuries of the Church already favored either T or + shaped. The Epistle of Barnabas (c AD 90-125) records
For [the Scripture] saith, “And Abraham circumcised ten, and eight, and three hundred men of his household.” What, then, was the knowledge given to him in this? Learn the eighteen first, and then the three hundred. The ten and the eight are thus denoted—Ten by Ι, and Eight by Η. You have [the initials of the, name of] Jesus. And because the cross was to express the grace [of our redemption] by the letter Τ, he says also, "Three Hundred." He signifies, therefore, Jesus by two letters, and the cross by one. (Barnabas 9:7)
In Greek numerals, 300 is written with a tau, resembling the Latin T. The author of Barnabas clearly thought the cross was shaped like the Greek letter tau, T. In Greek, Jesus is IHSOYS -- I and H being the initial letters.
A bit further in the epistle, he writes
Here again you have an intimation concerning the cross, and Him who should be crucified. Yet again He speaks of this in Moses, when Israel was attacked by strangers. And that He might remind them, when assailed, that it was on account of their sins they were delivered to death, the Spirit speaks to the heart of Moses, that he should make a figure of the cross, and of Him about to suffer thereon; for unless they put their trust in Him, they shall be overcome for ever. Moses therefore placed one weapon above another in the midst of the hill, and standing upon it, so as to be higher than all the people, he stretched forth his hands, and thus again Israel acquired the mastery. (Barnabas 12:2)
Thus Moses, with his arms outstretched is described as a type of Christ on the cross. This suggests a T- or +-shaped cross.
A little later (c AD 150), Justin Martyr wrote
God does not permit the lamb of the passover to be sacrificed in any other place than where His name was named; knowing that the days will come, after the suffering of Christ, when even the place in Jerusalem shall be given over to your enemies, and all the offerings, in short, shall cease; and that lamb which was commanded to be wholly roasted was a symbol of the suffering of the cross which Christ would undergo. For the lamb, which is roasted, is roasted and dressed up in the form of the cross. For one spit is transfixed right through from the lower parts up to the head, and one across the back, to which are attached the legs of the lamb. (Dialogue with Trypho, Chapter XL)
This seems to match with a +-shaped cross.
So it seems that the earliest Christians believed Christ was nailed to a T- or +-shaped cross rather than a |- or X-shaped cross.
If we consider what Matthew recorded,
And they put up over His head the accusation written against Him: THIS IS JESUS THE KING OF THE JEWS. (Matthew 27:37, NKJV)
which seems to rule out a T-shaped cross, the balance of the evidence favors the traditional Christian shape of the cross.