Why would a 5V input to a 74HC04 pull it's V_CC from 3.3V to about 4V
Chips' ESD protection circuitry typically works by directing any current that would pull an input above VDD, out the VDD line.  With its input tied to +5 volts via 1K resistor as shown, and VDD rigidly fixed at 3.3 volts, the 74HC00 would probably clamp its input to about 3.5 volts, and would take the 1.5mA flowing into the input and direct it out to VDD.  If other circuitry powered by VDD took 1.5mA or more, the redirected current would reduce the supply current draw by 1.5mA.  Otherwise, the current will have to go somewhere else.

Some regulators will perfectly happily sink current if their output voltage exceeds the proper level; others may be destroyed, and some simply won't care (when the output voltage is below the regulation point, they'll supply current, but if it's above they'll simply sit idle).  If you don't have something in your circuit which can absorb 1.5mA when an input is high, you'll have to add something.  The simplest such thing may be a 2.2K resistor from the input pin to ground; that will absorb about 1.5mA when the uC is outputting +5, while wasting nothing when it isn't.