I appreciate the effort, Ronin, but most of what you've said is incorrect in large or small ways. To go through it:
Iaido is indeed the art of drawing the sword, katana, or shinken more specifically referring to the "Samurai" sword
Well, "katana" (刀) really just means "sword" (although an argument can be made to a specific style, being a curved, single edged weapon), and "shinken" (真剣) means "real sword"... while both terms are used to describe Japanese style swords, neither really mean "samurai sword", and such terminology isn't common to hear aside from people completely removed from the topic. But I get what you're saying, and yes, Iaido is, ostensibly, the art of drawing a Japanese sword.
and performing cuts as well as defensive moves as in kata.
"As in kata"? If you mean that the teaching and training vehicle is kata, then yes.... the phrasing throws me off a bit, though, as it's done as kata, not as in (as if it was) kata... of course, what we're talking about here is Japanese kata... not karate (Okinawan-based) kata... which might be where some of the confusion comes from.
There are various styles with their own kata as in karate, as well as a series of standard kata agreed upon by most of the styles.
Hmm.... not really. Each separate art will have their own syllabus (the waza of Tamiya Ryu Iaido is different to that of Mugai Ryu, which is different to that of Muso Jikiden Eishin Ryu, which is different to the Iai syllabus fo Katori Shinto Ryu, which is different to the Iai/Batto syllabus of Kashima Shinryu, which is different to the Iai syllabus of Takenouchi Ryu, and so on...). There are a couple of more "universal" sets taught within their umbrella organisations, the most common and popular of which is the Seitei Gata of the ZNKR (we'll come back to that), but there are others, such as the Toho Gata, and, one might suggest, the Toyama Ryu Battodo schools.
Sticking with the Seitei Gata, it was indeed developed by bringing together teachers of various schools to try to come up with a "general" Iaido syllabus (we'll look at why later), such as Hoki Ryu, Mugai Ryu, Muso Jikiden Eishin Ryu, and Muso Shinden Ryu, but that is not saying that it is a truly "standardised" syllabus aired upon by anyone other than the members and leadership of the ZNKR.
The style-specific katas differ based on the style's lineage.
They differ based on the art... the lineage is obviously a part of that, but there's a lot more involved as well... such as context, how it fits into the larger syllabus (if not a stand-alone Iai system), and so on. In other words, Shojitsu Kenji Kataichi Ryu's methods of performing Iai in armour isn't so much a matter of their lineage, anymore than Katori Shinto Ryu's use of Tatehiza rather than Seiza is a part of theirs.
There is also tameshi-giri involving cutting soaked rolls of bamboo matting to simulate the resistance of a human limb.
The tatami-omote most often used is rice straw, not bamboo. In addition, not all schools do tameshigiri... those that do use it simply as a training tool itself, rather than it's own aim (the Toyama Ryu and Shinkendo are two of the few that treat it as a skill in and of itself).
Aikido is said to have been inspired by the cutting strokes of iaido;
No, not really. It's often suggested that kenjutsu played a part in the formulation of Ueshiba's Aikido, with the Yagyu Shinkage Ryu most often cited, although the connection to that school happened much later in Aikido's development, and had more influence on the Aiki-ken methods that Saito-Sensei taught. The biggest influence on the movements of Aikido taken from sword are more reflected in the footwork than the handwork as well, and is more likely a reflection of the sword taught alongside Daito Ryu that Ueshiba learnt from Takeda Sokaku (it was associated most strongly with Ono-ha Itto Ryu, being the official sword school taught alongside).
sweeping movements often returning upon the same line. This has the effect of capturing the attacker's movement and redirecting it back at him.
The idea of controlling the centre is common throughout many Japanese martial arts, so I don't really attribute that to any other factor other than "it works", and would have been encountered in each school Ueshiba studied. As far as "redirecting the movement back at the attacker", again, that's pretty much Daito Ryu 101, and nothing to do with sword in this sense.
As previously mentioned, if an Aikido dojo uses a "sword", most likely it's just to help visualize their empty handed techniques, although there may be an exception or two. Aikido dojos may teach empty hand defense against a sword attack.
Most Aikidojo will teach tachi-dori (sword capturing, literally... defences against a sword), whereas the actual use of sword will vary to a much greater degree. Most groups that have a larger focus on weaponry are the Iwama Ryu groups, stemming from Saito as mentioned above. Others use the sword, as you mentioned, to extend the lines and work on aspects of the unarmed work. Then again, there are a relatively large number of dojo that have separate sword approaches, typically brought in from outside, to work on those aspects, such as Inaba's version of Kashima sword work, the French Aikibudo group and their use of Katori Shinto Ryu methodologies, similar with the Yoseikan groups, others who incorporate a line of Iai or another.
Having spent just 2 years in Iaido, I'm not an expert, but from my knowledge and outside study, Iaido anid kendo are much different.
Well, yes... but that's not what Paul was talking about.
Paul was addressing the idea of training in both Kendo and Iaido simultaneously, as well as the idea of training in Kenjutsu and Iaijutsu simultaneously, and was quite correctly pointing out that Kendo and Seitei Iaido are designed specifically and purposefully to be trained together. In fact, Seitei Iaido was created, in large part, specifically to give modern Kendoka the experience and understanding of using an actual sword, as opposed to a shinai, which gives a very different physical response.
In other words, (modern) Kendo and (Seitei ZNKR) Iaido are complementary arts that are designed to be trained together. After all, the ZNKR is the All Japan Kendo Federation (Zen Nippon Kendo Renmei)....
Kendo is a competitive sport off-shoot, involving hits (touches in fencing) with a bamboo replica. Moves are short and quick for the point score.
That's one way of looking at Kendo.... I'd argue it's a lot more than that, but okay.
Iaido involves cutting which involves broad, sweeping moves for the most part and a different mindset.
That can depend greatly on the school... and doubly so for the mindset.
Musashi Miyamoto in his Book of Five Rings goes into this concept.
Considering modern Kendo and Iaido were developed some nearly 300 years after Musashi-sensei died, no, it doesn't.
It also involves a steel blade (sharpened in my dojo, other schools may use a blunted blade).
The use of a shinken is not so common for beginners.... in most dojo, you'd start with a bokuto (more for financial reasons as anything else), then move onto an iaito (typically made of a zinc/aluminium alloy that allows for the weight and balance approximating a real sword, but unable to take an edge), before much later (usually after 3rd Dan in Seitei groups) moving onto a shinken (live blade).
Kendo has no saya (scabbard) so there is not a true draw or resheathing of the sword.
Kendo is about the engagement with swords already drawn... same with kenjutsu... so no, drawing the sword (other than symbolically) is not part of the art... mind you, that's like saying that learning to be a pastry chef doesn't involve cooking a casserole, so it's lacking in cooking skills.... no, it's just not part of it's scope.
Both arts involve discipline and mental clarity and may share a few other things, but in performance look nothing alike. I feel like they should be considered as two completely different things - maybe similar to Judo vs Aki-Ju-Jitsu.
Leaving off the idea of Judo and Aikijujutsu there, the relationship between (modern) Kendo and (Seitei) Iaido is not anywhere near as separate... again, they are complementary... two aspects of the full study.