As I tried to say in the first post, you need to match your needs, your software, and your soundcard.
[A] If the recording application you want to use allows real time input monitoring/mixing, then you don't need that facility in the soundcard; you can get by with an external soundcard and no hardware mixer.
[B] If you want an external soundcard and want to use software that does not do real time input monitoring, you need
(1) a full duplex external soundcard (common) to play back the existing tracks through its DAC while recording the new ones through its ADC
(2) direct input monitoring in the external soundcard to monitor the new input that is going into the ADC
(3) a hardware mixer (separate device in most cases) to mix both (1) and (2) before sending the signal to the speakers.
[C] You can use a PCI soundcard, even a multi-channel one where the converters are in an external box. These have DSP mixing that lets you monitor the new input mixed with playback of existing tracks.
QUOTE
record instruments and incoming audio signals onto my computer (in a software mixer)
If you do multi-track recording, each source to a separate track, you will have a great deal more control over your final product. This can be done with a multi-channel soundcard, if you are creating several different things at the same instant, or with a basic two channel soundcard if you can get by inputting only one or two things at a time. Both approaches are common. Nothing gets mixed (in any permanent fashion) until everything is recorded and each part tweaked to be exactly what you want.