7-Segment Common Anode
You can use a single resistor on the common pin if you want. The problem is that lighting up multiple segments at the same time will substantially reduce the brightness of each one. You could work around that by lighting each segment one at a time in very quick succession though.

It should be possible to control the display directly from the IO pins by setting them LOW. In this configuration, they will be sinking current (rather than sourcing it). It depends on the specification of the display and on what resistor you use though.

Basically, try to keep the current below 20mA and it should be OK (assuming you're using a standard 5V Arduino). For example, if each segment has a forward voltage drop of 2V, you would want to use a resistor of around 180 Ohm.

Controlling the display directly from a shift register is probably not possible, although once again it depends on its specification. Shift registers usually aren't designed to sink enough current so you'll need something like a transistor on each output.