Can you use AIC/BIC to compare models on untransformed data with models on transformed data (such as log, inverse hyperbolic sine, etc.)? I.e. if a model using logged data gives an AIC = 53.62 and a model using the the non-logged data gives an AIC = 150.6, can you say that the logged model is better even though the scale of the data is different between the two models?
Your question appears to be a special case of this post which explicitly considers prerequisites for AIC model comparison. One of the important prerequisites is that the dependent variable (the vector of observations) is exactly the same across the models being compared; see the answer by @usεr11852, especially his point 2. The dependent variable cannot be a transformation and it cannot have different number of observations in different models (which may effectively happen, for example, when comparing ARIMA and exponential smoothing models estimated on exactly the same set of data (!); read carefully the brief answer to this post).
However, transformed dependent variables can in principle be handled by adjusting the AIC for the transformation: see a comment by @probabilityislogic under his/her answer here and the comment by @CagdasOzgenc to your own post.
You may find a number of relevant posts arriving to similar conclusions in this list. Perhaps some of them will be formulated more clearly than I have managed.
No. The absolute value of AIC indicate nothing. The residuals are included in AIC calculation, so when you transform your data your residuals will be "transformed" too (actually, you will calculate new residuals for transformed data, and they are incomparable with your initial residuals).
It can be seen, i.e., from this guide, chapter "MINIMIZING THE LOSS OF INFORMATION".