after my engine is warm and I go to restart it, it runs lean (~17:1) and idle is lower than normal (~500rpm). I have to tap the gas a couple times to get it in check, or if I just drive it usually by the next time I come to a stop the problem is gone. It's usually only on hot starts, but sometimes it randomly happens when i come to a stop as well.

Is there anything electrical I should check? It seems like when I put on a lot of accessories at idle (heater, defrost, music) the idle drops and it gets leaner like it wants to die. I've replaced the alternator, it didn't help. Battery is from Honda one year old. What should I do? problem has been going on for months. I flushed out the coolant system last week, this didn't change anything either.

 

bump
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Who did you tuning? Have you contacted them yet by chance? May be an issue with the tune I am assuming. This is a long shot, but how do your engine grounds look?

 

Going to take a shot and say Idle Air Control Valve. Mine is pretty much shot and has similar but worse issues. You might get by with a good cleaning of it.
Search IACV here and you'll find some good write-ups and other threads with similar symptoms.