Using the Inverse Filter to Correct a Spatially Convolved Image (Deconvolution)
There's a couple subquestions that I'll address separately:

Convolution in the spatial domain (or correspondingly in the time domain for time-sampled signals) is equivalent to multiplication in the frequency domain. In sampled systems, there are some subtleties to boundary cases (i.e. when using the DFT, multiplication in the frequency domain actually gives you circular convolution, not linear convolution), but in general, it really is that simple.

Pure inverse filtering is almost never the right solution in practice. In most cases, you don't have access to the exact filter that has been applied to your data, so you can't simply invert it anyway. Even if you do know the filter, then it's still problematic. Consider the fact that the filter may have zeros at certain spatial frequencies; if it does, then after applying the filter to your image, all information at those frequencies is lost. If you naively invert that filter, it will have infinite (or at least very high) gain at those nulls. If you then apply the naive inverse to an image that has any additive content at those frequencies (e.g. noise, which is likely to be the case), then that unwatned component will be greatly amplified. This is generally not desirable.
This inverse-filtering problem is very similar to equalization in communications systems, where this phenomenon is referred to as noise enhancement. In that context, the inverse-filter approach is referred to as a zero-forcing equalizer, which is rarely actually used.

The area that you're exploring is known more generally as deconvolution. As a general rule, deconvolution is a tricky operation. Even if you know the exact filter that was applied and want to undo it, it's not always that easy. As you noted, the inverse filter approach is usually brushed aside in favor of a Wiener filter or some other structure that aims not to exactly invert the system, but instead to estimate what the input to the system was while minimizing some error criterion (minimizing mean-squared error is a common goal). As you might expect, applying a Wiener filter to this problem is referred to as Wiener deconvolution.