When summoning tamed animals there is a delay after dismissing before another one can be summoned. This is several minutes of delay to e.g. train the summon skill. However there is no delay summoning rewards (e.g. the Chest Mimic). This can be dismissed and summoned again immediately. Doesn´t that defeat the purpose of having a delay to summoning and training the summon skill? Edit: And I managed to tame a Large Brown Spider - only to notice that I could not summon her due to a too low summon skill... Should that not be unison for both skills? e.g. if an animal is too high level to be summoned, then it should be too high level to be tamed, too. Summoning an already captured and tamed animal should always be possible - or taming not possible.
Does the chest mimic give you experience in your summoning skill? If not, that's the entire reason. Tamed pets have a timer that can be countered by the refresh innate skill that can reduce this timer down to 0 at lvl120. A chest mimic is neither affected by the taming skills or effects, so it makes sense that it doesn't afflict you with tamed pets debuffs either.
Game systems are not necessarily envisioned to make sense in a real-life way, they are usually designed to make sense in a game-balance way. The taming tree is ultimately the most powerful PvE tree in the game and if you want that power then you have to put in the time and effort. With that said, each time you use a skill, some portion of your experience pool gets put into that skill. The larger your experience pool, the more experience gets pumped into a skill with each use. In order to raise the Summon skill faster, lock all your other skills and allow your experience pool to grow. Also, some food items, the obsidian potion of guidance and the blessing in Atos Castle can help. There's also a blessing in The Lost Vale but that might be beyond your reach right now.
there is the one founder mimic that is combat which is affected by taming but its capped at level so it does not get powerful beyond its level like the other tamings.