Tapping or pushing the button fully shouldn't make a difference as far as I know.

If this is a new bike I would definitely ask the LBS to take a look, this shouldn't happen.

The remaining battery charge doesn't matter in this case - either the FD shifts or it doesn't. Once the battery charge reaches 10% or so (not sure about the percentage) the FD will stop shifting completely until you charge the battery.

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Thank you.

Which model FD? The newer models don't have an inner limit screw; they use a soft inner limit that you program manually. It's possible you went into this mode and messed it up. Read the installation manual (online).