Is there a minimum for charging current for lead acid battery?
The usual rule for charging a flooded lead-acid battery is that the charge current should be less than 20 - 25% of the Ah rating.  for your 4 Ah (4000 mAh) battery,. that would mean a maximum charge rate of about 1 Amp.  Gel and AGM batteries can accept a higher charge rate.

I don't think there is a minimum charge rate, but the lower the charging current, the longer it will take to charge the battery.