Its the actual AutoDJ files that you have uploaded that you need to address here. Either the mixes have been recorded too loud in the first place, or they have been hammered with compression and hard limiting in post processing. As a music producer of 14 years myself, I can hear that they have been pushed quite hard in order to achieve maximum loudness.
I had a listen myself and also ran the stream through an Orban meter (these are an excellent piece of freeware; you do need a Win7 machine for the latest version but it doesn't need to be top of the range).
Your loudness level is about -9 LUFS which is loud, but normal for youth/EDM stations (although EBU suggest -14 for these and -23 for classical music/speech based stations).
The "high volume on start" issue could be caused by listeners on some Android mobile devices using headphones. I have heard it myself on other online stations; but always on one device (a rather battle scarred old HTC one with Cyanogen mod). It appears to be a side effect of some hardware/software methods presumably used to save battery power.
your station sound does appear to have been "double processed".
Easy to do by accident if both your DJ software and playout software has soundprocessing plugins and you end up sending an already processed recorded mix into ingest and playout; or you overdo it with any software that is clamed to make your source content (music) sound "louder".
Another pitfall is level meters on hybrid analogue/digital kit (or on screen simulations on them) are often wrong scaled and encourage redlining (we are not using spools or cassettes of tape any more!). With online radio the encoding/decoding process of the stream can push the reconstructed peak level of a loud signal beyond 0dBFS digital.
It is then dependent on the listeners software and equipment what happens; the best case is the sound card has some headroom or it hits a limiter
before its converted to analogue; but it could equally just put out a harsh distorted sound; not good for speakers nor ears. You can get away with very short occasions of this but do not want large amounts of the programme audio to be at this level.