The question of what makes an 'accurate' reproduction is hard, and it involves a lot of nuances that are hard to convey in a post here on r/askhistorians - particularly since this is not a forum geared towards reenacting, but towards the original events (and material culture and beliefs etc.) of the past. Among other things, it is hard to say how accurate a reproduction is just by looking at it on the internet - a lot of authenticity is things like the weight of metal it is made of, the exact construction of the rivets, things like that.

That said, this looks like a mass-produced 'costume quality' helmet. It is made of rather thin metal from the looks of it - original helmets are one of the few pieces of armour that are really thick; they can weigh up to five pounds or more. This gives original helmets and exceptional reproductions a kind of 'solid' look that I don't see here. The finish of this helmet has a kind of machine-made look - not that originals where shoddily made, but all but the highest end original helmets have the slight irregularities of handmade goods - slight asymmetries, hammer marks in places that don't show, that kind of thing. The Brass applique, the rivets (which are all identical, so have a 'modern' look), the placement of the breath holes are all very precise. The biggest construction issue is that it looks like the top of the skull is not a separate piece. In original 13th century great helms (which in other ways this helmet resembles) the top of the skull is generally a second metal plate rivetted onto the sides, like this example from the second half of the 13th century in germany. The brass applique is generally based on this example now in the Germanisches Nationalmuseum in Nuremberg. Note that in the original the aplique forms a much finer fleur de lis pattern rather than a simple maltese cross. It is also a 14th century great helm, not a 13th century great helm.

The combination of characteristics of helms from slightly different eras, the 'machined-looking' finish, thin material and simplified construction are what I mean when I say that this is a 'costume quality' helmet. This helmet would be great for cosplay or plays or another kind of costume, but I wouldn't call it an authentic replica. But this gets at a bigger question - what do you want this replica armor for? Are you doing public education (in which case authenticity matters a lot)? Do you want to wear it to a con or Rennfaire? Do you just want something that's cool? It might be that you don't need an authentic replica for what you're doing.

Authenticity is difficult and expensive - people that dedicate themselves to this spend years researching originals, and their work is never done - the bar is always rising. To give you an idea, the museum curator Tobias Capwell wrote his doctoral thesis on the armour of English men at arms, referring to hundreds of funerary effigies (statues depicting the deceased in churches). To test out his ideas, he commissioned a harness of plate armour from one of the world's greatest living armourers, Robert McPherson, and collaborated on its design and construction for well over a year. Now, not everyone is going to do that, but that is one definition of 'accurate.'

But anyway, back to the historical inspiration behind this piece. This general style of great helm would be worn in the second half of 13th century . This is after the most famous crusades - Richard the Lioneheart and Saladin were both dead for decades before this kind of full great helm came into its own (earlier helms only covered the face, not the back of the neck). So crusaders on later Crusades like the 7th or 8th or 9th might wear it, but not crusaders on the third crusade, and certainly not crusaders on the first.

This would be worn with a mail hauberk (shirt) with long sleeves, that extended to around the mid thigh. Generally the sleeves would include integral mail mittens, called mufflers. The very often had a mail coif (hood) built into it. In the 13th century knights generally wore mail chausses (leggings). There are references to a rigid torso defense called a cuirie, which was probably made of hardened leather. Early versions of the coat of plates (a form of plate armour where medium-sized metal plates are rivetted to a backing) were increasingly worn in the second half of the 13th century, and were common (based on documentary evidence) by the end of the century. Later on in the century plate armour supplements might be worn over the knees, elbows or shoulders. I write more about 13th century plate armour here. Over all of this a knight would generally wear a surcoat - a loose, flowing garment (that happens to obscure whatever is protecting his torso from the gaze of later historians). You can see armour from the later 13th century in the brass of Roger de Trumpington. Note that by this point (the last couple decades of the 13th century) some great helms have a conical top - his also seems to have a brass decoration in a fleur-de-lis design like the surviving German example mentioned above.

We also sometimes see what might be hardened leather armour worn over mail on the limbs. Keep in mind that leather armour was generally a supplement for the primary defense - mail.

Regarding mail, it was generally 'woven' in a 4-in-1 pattern (each ring has four others linked to it). At least one 14th century mail neck defense uses a 6-in-1 pattern for greater strength - often mail defenses for the neck and other vulnerable areas would be made of rings with smaller diamater and thicker metal, at least in the 14th and 15th centuries (very little 13th century mail survives). In the 13th century it was often made of alternating solid and riveted rings. 'Butted' mail - mail closed with pliers - was very, very uncommon, to the point where I have found a single written reference to it (in a Tower of London inventory from the 14th century, as reported by Thom Richardson). Mail would be 'tailored' to the wearer - when you look at depictions of knights in mail their mail often has a very 'fitted' look - it is important not to add weight by wearing a baggy hauberk! Padding would be worn under mail, but it wouldn't be massively thick - thicker fabric defenses were generally worn -over- mail. Again this saves weight, because less area needs to be covered with mail.

Please follow up with any questions you have.

Sources:

Alan Williams - The Knight and the Blast Furnace

Thom Richardson - The medieval inventories of the Tower armouries 1320–1410

Edge and Paddock - Arms and Armour of the Medieval Knight - this is a good introduction on knights' armour and weapons from the 11th through the 16th centuries, and is available at many public libraries.

Claude Blair - European Armour C. 1066 to C.1700