Whats the difference between installation of various devices?.. I think theres nothing different... Just put the installation package (u´ve downloaded from MIUI website) into your SD card, get to the HBOOT using volume dn&power buttons, then choose recovery, Install zip from SD card and navigate...
SD Card has nothing with ur custom ROM storage... You dont flash the ROM into the SD card like OS Windows into HDD.. Its only "increased partition" of ur phone storage so you can exchange ur SD card without saving any data besides ur apps/photos/any other files..
I dont think that its software problem...battery problem? try to charge ur device with another battery.. and at last like u said, ur device died? I cant say that definately but I hope that its false :/
And does the connector goes in correctly? I mean, dont you have any problems with connecting the cable into the port?
Another idea, If you look inside of the port, theres small 5mm long board in the center of it. Doesnt it look damaged or broken?