make loud noises quieter

If it’s multi channel surround-sound there will be a low frequency bassy channel, including that channel in your mix at the same level as the other channels may explain why bassy sounds are extremely loud in your mix. You can use the level slider (I’ve marked it green on your image below) to adjust the level of each track in the mix. Cut back the level of the dog-bark door-slam channel before saving the mix as MP3.

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You’ve cut off the time scale, so we can’t judge the attack from the image.
However the amplitude scale is shown and parts of the waveform are beyond +/- 1.0 (it shouldn’t do that).

If you use Audacity’s “Amplify” (in “Effect” menu) on default settings, it will reduce (sic) the amplitude of the waveform so none of it is beyond +/- 1.0 , (the red lines I’ve drawn on your image). .

Then you could apply dynamic range compression to make the loud bits quieter and quiet bits louder.
Chris’s dynamic range compressor is better then the compressor which is shipped with Audacity.