Apparently high tier armor is locked behind clan tier, so once you get to a higher tier, the higher tier armor 'should' start showing up in shops. What tier though? I'm not 100% sure since I haven't checked the files myself, but from what I've gathered, it should tier 5 or 6.
There are 50 achievements in Mount and Blade 2 Bannerlord. On this page, you will find a list of all achievements along with descriptions on how to get them. Against all odds. Apple of my eye. Bannerlord. Best served cold. Butcher of Calradia. Butterlord. Catch.
Yes they do. I have increased in clan tier when they have defeated an enemy before. From what I can tell they: Give income after winning battles. Give renown after winning. Give influence after winning. Increase relations when helping other parties. < >
Fastest way to upgrade leadership in my experience is to lead armies. Join a faction, level up influence, and then just create armies (you can recruit other existing lords or do the thing where you turn companions into their own party and constantly recruit themor both). Leadership goes up pretty rapidly the larger the army is that you lead.

Sometimes, the lords will be willing to pay you to join your faction. Other times (more common), you'll need to pay anywhere from 100k to over 2 million. This can be circumvented by recruiting lords with the Honor trait or by recruiting lords with which you have >30 relation (as of 1.4.0). These lords will join you for free.

If you want to join in as a vassal, you should first increase your clan tier and reach the required tier to become a lord. So, the best way you can join a kingdom in Bannerlord would be to: Gather enough Renown to reach Clan Tier 1 or 2. Approach either a lord of the kingdom (to become a mercenary) or the faction leader (to become a vassal).

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mount and blade 2 bannerlord how to increase clan tier