From what I can see, 8 bit AVRs aren't fast enough to play MP3. Instead projects mostly rely on decoder chips. The idea is that the decoder will decode the MP3 in hardware and produce output signal which can be later processed in an amplifier and sent to a speaker.

From what I can see, you'd either need to get a board which can do the decoding for you like this (this one has SD card reader, amplifier and a small speaker too) or this (both use VS1011E decoder) or make your own board which will house the decoder, amplifier, SD card reader and so on. Here you can find a project which explains how to do that and here is one which used AVR Butterfly platform. Both projects have screens and are battery powered from what I can see.

Let's do some math:

16 tracks * 44 ksample/s *2 channels *16 bit = 22.528 Mbps

This is the minimum speed you need for the SPI interface, if you want to transmit all the data through a single serial port. Can be done, with an adequate clock, but you need a fast SD card (see here for the speed).

Then there is the microcontroller: you have to add 16 tracks and output them through a DAC, so you have 44*2 ksamples for each track, or

$$ 44 \cdot 10^{3} \cdot 2 \cdot 16 = 1408 \cdot 10^{3} $$

16-bit sums for every sample (probably with some scaling to avoid overflow) result in about 1.4 M operations/sec, that can be handled by a good 32-bit microcontroller. Probably a Cortex-M3, or better M4 (but M3 is probably better documented) can work for you.

I've just seen this which can be clocked up to 204 MHz, has 4 SPI interfaces, up to 40 MB/s, and has also a floating point unit that can help in the accumulation process (but may be too slow). You may also use the dual core structure to handle separately the processing and the output.

But for the DAC I think that you should go for an external converter, specifically designed for audio (which means 16 bit probably).

Update

It's not so clear how are you going to manage the 16 different tracks on the SD:

• what about pre-loading tracks on the internal memory of the MCU?

Check the I²S interface, which is a 4-wire serial protocol especially designed for audio applications.

Important question:

You said that you want also to record tracks and save them to the SD card: do you want to do that at the same time? You need the controller to encode the audio in WAV and store it, and the writing bandwith of the SD card is lower.

The looping feature WILL need some buffering memory (may also use the internal memory) because looping requires real time operation, and the SD card will introduce too much latency. You may need an external RAM, and you may also think about storing some data there to reduce delays.