The most challenging maps for you.

To build a treasury, you need an extra 3 days and 17,500 gold, but it only pays for itself after 11 days. Of course, if there's plenty of free money on the map, then you can take your time. But I haven't noticed any extra cash lying around on the road without protection, especially at the beginning of the game. However, I have noticed that sometimes guests might show up after a week, and it wouldn't be very dignified to receive them without an army. So, in short, build a treasury.

I don't know what a treasury is. I meant that first you should build a magistrate and something else before it; I don't remember what it's called in Heroes 3 anymore. Because I've played all versions – 1, 2, 3, 4, 5 – on impossible difficulty. And it doesn't cost 17,500, but rather about 2,500 + 5,000, and you can build a capitol later, after building troops.
Sincerely.

Added 6 minutes ago

I would add: of course, much depends on the map. If the map is large and enemies cannot come because of border gates or guards (in cases where the author intended only you to be able to visit enemy territory), then you can start with a capitol. But if the map is ordinary, how will you fight enemies without an army, even if you have a billion gold? :confused:
As for playing as Castle, especially on impossible difficulty: I would like to see that! I was going to write about why Castle is one of my least favorite cities. There's a topic about this; I don't think it has been discussed yet... Here, I wrote it: /topic/8304/

What's so difficult about Castle? Here's a simple strategy for playing as Castle. 1) To build initial and mid-tier units, you only need wood and stone, which are plentiful everywhere. You should launch 2-3 heroes at the beginning. You won't drag things out for long; by week 2 or 3, you attack an enemy castle, preferably two, doubling or tripling your income. Moreover, I try to attack specifically enemy "Castles" so that my troops get stronger. But the main principle is to start quickly; everything else can be taken from the opponent: Titan castles, for example. By the way, I easily defeated Titan armies. And without any magic. I don't even use autobattle; I choose the option to show "battle results." And if you control the battle manually, you can completely crush them into dust. Moreover, I don't use any bugs, cheats, or other nonsense like hero suicide or buying a new hero in Castle. Otherwise, it's not interesting to play; it's like boxing against a 5-year-old child. By the way, it's best to play the Champions of Gelanor scenario with Castle; I find it easier. Read online about Castle's capabilities. I learned about the abilities of upgraded bull gorgons, wyverns, and dragonflies only after playing as Castle on impossible difficulty. It's actually quite easy there. Usually, I go with half my army, not yet upgraded, to the enemy castle, defeat it, and only then do I see upgraded gorgons. Only later did I learn that basilisks slow down the army during movement. And that hydras are useless in the general army; they should be used individually. IMHO.
You better tell me if you played the Heroes 5 scenario "Days of Flame" on "hero" level. That is really difficult. And here too, I started with money-making buildings. If you're interested, I can tell you how to play it. I completed it on recruit level in 21 days, and on hero level in 71 days.
Sincerely.