I'm no expert on the history of Italian medieval warfare, but I'm fairly certain that many of the items shown in this thread only appeared after the suggested time-frame for this mod.
-The bascinet only evolved in the 14th century, http://www.myarmoury.com/feature_spot_bascinet.html
while the barbutta and sallet (salade) were 15th century helms.
-There may have been halberds in the 13th century, but I have yet to find any specific references to them (again, not that I've spent a ton of time researching that)- though I see Angorod has said this too
WEAPONS: Spear and sword are still the universal weapons of the mounted knight, supplemented by one-handed maces or axes. Crossbowmen are common (Northern Italy exports them as mercenaries). Infantry spearmen are common, as are a wide variety of staff weapons (battle axe, ahlespeiss, glaive, maul, and guisarme). The falchion (a heavy-bladed sword, effective at cutting through armor) appears in this Century and remains popular among noble and common soldier alike for several centuries.http://surbrook.devermore.net/herosource/armoroverview.html
-plate protection, such as cuisses, cuirasses, spaulders, gorgets, etc. did not exist in 1257. Poleyns did exist, though they would have been uncommon, as well as shynbalds and couters.
-the types of padded jacks shown are also more like the 14th century, I think. Here's a description of 13th century gambesons:
Long gambeson, patterned on italian iconography. Fastened with 2 buttons, quilted torso (horizontal zigzags), the lower part engraved and decorated with stripes of fabric in different color.
Gambesons were a cheap protection for the fighting men in the Middle Ages, but only the poorest ones used only them as their single armour. For others they were a part of a complete protective armament, providing amortization of the hit and the protection of attritions from the armour. Among padded protective elements we can find: gambesons protecting the trunk and hands, padded gloves, padded coifs and separate hoses. They were sewn from layers of a fabric. In the 13th century gambesons were long and rather badly-fitted to the figure. They were usually put over the head and buttoned only near the neck. Later, with the development of other elements of the armour, gambesons were shorter and better fitted to the body. In the 14th and the 15th centuries they were usually buttoned at the front or laced up.http://www.medieval-market.com/details.php?id_towar=162&s=1
General list of 13th century equipment: http://cunnan.sca.org.au/wiki/13th_century_fighting_equipment
Anyway, if we're prepared to ignore the time-period inaccuracies, I'm prepared to make some cool helms and armour. If not, well... the Norman style helms are really cool too.
*I also posted this in the research thread (I think), not sure which is more appropriate