Yes, it is actually not that hard to use an AVR chip on its own. You only need to take care of a few things and it will happily do its work. Most AVR chips aren't fussy about the conditions either, so there's quite a bit of room for error. You can make it as simple or complex as you want. Adding a JTAG connector is probably the easiest way to program the chip in your own circuit. The Arduino board has one*, so you can see how that works and figure out the wiring before you make your own attempt.
If you build a board with a JTAG connector, an oscillator circuit consisting of just one crystal and two capacitors, one or two decoupling capacitors and a linear regulator, you basically have an Arduino without the USB capabilities. That sounds like quit a list, but is actually a small number of very doable tasks. The best part is that you get to use the IDE you already know, and the Arduino can be used for testing and as an example. When you get more comfortable with how things work, you can migrate to Atmel Studio for some proper low level control and efficiency. AVR chips are simple enough to understand, but complex enough to do some very interesting things.
One thing to note is that the choice of microcontroller in a production product depends on many things, amongst which your experience with various microcontroller families, cost, power consumption, processing power, licensing and more. However, that tends to be less critical in smaller production runs. If you don't launch a product into a mainstream market with razor thin margins, any chip that gets the job done and doesn't have some prohibitive property, like excessive cost, should do fine.
*Technically the Arduino board has two, but only one for the ATmega328P.