Understanding the DCT even symmetry
Just as the DFT assumes a periodic signal by construction, the DCT assumes an even signal by construction. We can think of taking a DFT of a non-periodic signal by extending it periodically, but restricting our "viewing window" to the original signal itself. Now, what happens when you take the DFT of an even signal? The DFT is then real (which means it can be constructed with only cosines as basis functions). So, in a similar spirit to the periodic extension, we can take an arbitrary non-periodic, non-even signal, mirror it, then extend it periodically. This results in a real, cosine-based DFT. Again, we restrict our "viewing window" to the original signal itself. Note that although we've doubled the length in the frequency domain, we only have real components. Effectively this means we store the same amount of information compared with the traditional DFT. The reason the DCT is useful in compression is that compared with the traditional DFT, there is more energy in the lower frequency components than in the higher frequency components (energy compaction), so we can compress those higher frequency components. This is true because, as you said, this extended mirrored periodic signal lacks the sharp discontinuities of the traditional DFT input and as a result is smoother.