Regarding 7447 7-segment driver:
Do I need resistors here? Where would I connect them?
I've never seen this chip used without current limiting resistors, when driving LED seven-segment display. Generally, each of a,b,c,d,e,f,g
output pins gets a similar-value series resistor to limit current.
simulate this circuit – Schematic created using CircuitLab
Since this chip wants a +5v DC supply, LED common-anode pin is often connected to +5V as well. This is convenient, but not required. Each LED of a common-anode array needs at least a few volts to even begin to light up. The 7447 internal transistor adds a few hundred millivolts to this, leaving perhaps 2.8V for each series resistor to soak-up.
If you decide that 5mA LED current provides a bright-enough display, then each of seven series resistors would be 2.8V/5mA = 560 ohms. Some LEDs are super-bright while others (perhaps older) are not. Brilliance is adjusted by substituting other resistor values - brilliance is almost proportional to current.
The 4511 seven-segment driver is a different animal, not only because it is meant for common-cathode displays, but because it is based on hybrid CMOS/BJT logic. But the same current-limiting method is used to drive LED displays (from a data sheet acquired from Harris Semiconductor SCHS072B - Revised July 2003):
Extra question: Could I use a NOT gate on the common anode to make the
inputs "1" turn the LEDs on?
Inverter at the four 7447 BCD logic inputs? Simply - No!
Seven inverters at all seven 7447 outputs? Possibly, but a well-engineered circuit would include seven pull-up resistors to +5v at 7447 output pins.
You couldn't drive a common-anode LED display from inverter outputs, but you could drive a common-cathode LED display through seven series resistors from inverter outputs. A many-parts, awkward solution. Many TTL inverters only weakly provide drive current to a common-cathode LED display. You should carefully search for an inverter meant for high-side driving. The 4511 is designed for good high-side drive.
Common anode seems simpler circuit wise
... the common anode/cathode part of the circuit is the same in both the circuits ... the resistors in the second diagram are related to the input of the driver chip, and are not related to the display type \$\endgroup\$