That's a lot of questions. Here goes.
The biggest question right now is the amount of supplies and equipment we think we'll need. There's four stages I can think of basically. 1What we carry on our backs and horses=> 2A supply horse/mule or two => 3we have a cart or carriage=> 4so our cover is that we're part of a trade caravan.
I'm thinking 2 or 3 (leaning towards 2) for a nice balance of contingency equipment/extra supplies and maneuverability.
Aim for an amount that can be carried in the saddlebags of three horses. Horses are expensive, so it's two extra horses for you companions or a mule to carry all of your stuff for now.
Is it possible to get a topographical map and/or political map of our nation so we have an idea of the terrain we may end up traveling?
For a topographical map you can just take one from Japan. Do note that I might change stuff around if I feel it's better for the story (or if I don't feel like bringing up a map every time you move somewhere), like the fact that Nara doesn't actually border water in real life, but I'm extending the territory a bit southward so that it does because I feel like it would be something the scorpion clan would want to have. Expect lots of mountains and forest, with some plains thrown in. As for political, there's the little map in the second post. Underneath that you've got each of the great clans with their base province next to them. You may assume that the base province is completely under their control, the provinces around that one are mostly under their control, and anything further than that they'll have a dubious hold on. Mantis clan has the entirety of Shikoku island. Dragon clan have a hold on half of Kyushu, with influence on the rest of it as well. Etc. If you want I could probably make a map with pretty colours by spending 1000 hours in paint. Let me know if that's something you all would think is handy.
Who was the last Shogun and when did they die? Did we work for them, the current Shogun, or both?
The previous Shogun was Crab clan, Fujimori Benjiro. A pragmatic man who thought that war and bloodshed gave little profit in the long run, but who wasn't as naive as to not be prepared for it. He mostly used diplomacy where available, an iron fist where not, and tended to rub politicians and nobles the wrong way due to his rough demeanor. He died of natural causes at the ripe old age of 67, after ruling for about 25 odd years. Hideki never worked for him, as he died when Hideki was still working for the Scorpion clan. It was only a few years after the the new Crane clan shogun got into power that Hideki started working as the Shogun's Shadow.
Can we get a basic chart of the relationships between the Great Clans as we know them? I'm trying to draw probable battle lines in my head and have realized that I know very little about the politics of this world. Right now the best I have is Phoenix/Lion/Crane vs Scorpion/Mantis; MIGHT be in favor of Scorpion: Crab, Dragon; Wildcards: Spider, Wolf.
Pretty much. The Phoenix/Lion/Crane form a big power bloc. They're against the Scorpion/Mantis for their blatant relations with the Nanban and the fact that these two "lesser" clans still hold high positions in the bakufu/imperial bureaucracy.
Individually, the Lions have a rivalry going with the Crab clan, because they feel like the Crabs don't follow honourable Bushido. They like the Phoenix and Cranes.
The Phoenixes dislike the Dragon because the Dragons use magics other than that of the heavenly Kami. They also dislike the Spiders because they are said to dabble with demons and dark magics. Friends with Lion/Crane.
The Mantis are on the Scorpion Clan's side, both due to shared itnerests and family relations. They poke fun at the Cranes for their uptightness, but there's no actual hostility yet.
The Crabs have a rivalry going with the Lions, because they feel like the Lions get so hung up on their codes that they don't make battle in an efficient way. They're wary of the Spiders across their coastal borders, but no hostilities have been had and diplomacy is under way.
The Cranes like the Lions and Phoenixes. They dislike the Mantis for their frivolous ways, the Scorpions because of their high position despite being a smaller clan and the Spiders because they simply are the new kids on the block.
The Spiders don't really have a big problem with anyone, mostly because they stick to their home turf of Hokkaido and rarely leave it. They're partial to the Crabs because the Crabs do trade with them and show them some respect even though they are small.
In terms of size/power:
Lions have the biggest traditional army of professional soldiers.
Phoenixes have the largest amount of seers/priests who can use magic.
Cranes have a large army, but only part of it are professional soldiers. They have the most political pull and money. Rumours are that they're working on their own shinobi school.
Crabs have the second largest army of professional soldiers, just behind the Lions. Their soldiers are less well equipped and trained, but don't follow the honour code of the Lions which makes them less predictable.
Dragons have the second highest amount of magic users, lagging behind the Phoenixes. Their use more types of magic, however.
Mantis have pretty strong sway in politics, but nothing near the Crane clan's sway. They are not to underestimated however. They are also the best traders out of all of the clans, and their relations with the Nanban might shift power around quite a bit.
Scorpion, finally, don't have all that much of a regular army. They are the number 1 in stealth, infiltration and sabotage, however.
Finally, the Wolf clan is the top dog. Though they are small in actual numbers, they technically have the allegiance of every other clan, and all daimyo's answer to the emperor. To go against his word is to rebel against the Great Emperor of Jade and the Godess of the Sun themselves. In reality, they mostly keep out of things that happen outside of Kyoto. They aren't to be taken lightly though, because the emperor is said to be the strongest magic user in all of Nippon, and his retinue of Imperial Guard are the best warrior Nippon has to offer. Like the real emperor of ancient Japan, his power is technically absolute, but the day-to-day power lies completely with the Shogun.
I hope I'm making sense. You can always feel free to ask extra questions.