Sorry I wasn't able to check up on this all earlier; you guys posted right after I went to sleep and just before I woke up respectively. Now, to address your expectations:
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have story events to develop at a weekly basis
So, as I said in the initial recruitment post, the game will start with a little dungeon delve and then open into a more open experience to allow for exploration. I'll throw in quests and plot hooks as they come up naturally, but since this is a text game things will always come a bit slower so it'll all depend on how often people post responses and what life's throwing at people. This is something that I'd also like to have happen though since it's easier to stay engaged when there's stuff happening beyond trying to talk to a guard for a week.
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I expect world lore to be supplemental
I will admit that I am a heavier world builder (all of my games take place in the same world and thus affect each other) but it isn't a requirement to read all of my lore (especially since this continent is more removed from the rest of the world so a lot of what I have written isn't particularly applicable. I have many players who don't read all my lore and they function just fine because they'll get the info they need when they need it. However, skimming through things now and then isn't a bad idea either as it'll let you find cool little things that you might want to have your character do/see/obtain or whatever.
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I expect most of DM posts to describe a new challenge party has faced or narrate results of party response to previous challenge (or both in case narrative is that previous challenge is removed/solved/eliminated/escaped/etc.)
Now, I don't necessarily know exactly what this is meaning, so apologies if I don't answer what you were looking for. I will likely always start off a new 'scene'/area out with a description, allow for player interaction, and then respond based on that.
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I expect players to address challenge with every post, while adding role-play for flavor.
I will admit I'm not super well-versed in pbp gameplay, but I'm unsure on this one. Sometimes talking to a person or traveling is just part of the game and there won't necessarily be a 'challenge.' Obviously, some of this does depend on the players as well, but just since I'm not a veteran pbp gamer I don't know how realistic it is to expect constant threats/challenges/obstacles but I can certainly try.
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I wish our party could start the journey July real time.
That's my hope too! Just waiting for one more character and then we'll start!
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I wish DM narrate our PCs have already met at wizard tower and wizard explained us the mission and the contract and we begin with party going on the mission and facing their first challenge
I'll admit I intended on having a slightly different beginning (roleplay at wizard tower to ease in and then go onto the mission after discussion), but if the group would rather I can just narrate that out and start you off either on en route to the dungeon or at the entrance.
With time he discover powers and abilities different from magical. But there is none to teach him how to use those powers. He discovers his non-magical, but unnatural abilities could help him with anything he attempt to accomplish, it just take time and practice to use it. 
