Im not running puppy linux, i have used it before but mostly on USB keys in the past so I cant give much specific insight for this distro. You can try a audio mixer called pavucontrol not sure if that will help. There is more then one option for controlling audio, most use ALSA. Ive never had a linux machine leave the speakers on if another analog device was plugged in, headphones etc.
also you should be able to see if your SDcard hardware with the lsmod and dmesg command dmesg | grep card or MMC in Terminal. If the hardware is there, it might not be. you may have to mount the SD card manually if an usual format your system is not setup for automatically, ( mount commands or a mounting program graphically) althou it should just appear on your desktop just like a external hard drive would.
If you continue to have problems with puppy linux you can also try Linux Mint -MATE version (its what i use). Or the Mint XFCE edition, a more light weight desktop environment version. This or another distro might work better with your hardware.
You can make a bootable USB key using a program called unetbootin.
( better then using optical drive)
Hope this info helps, or at least points you in a direction leading to a solution.