You toggle between access of FLASH or EEPROM via the 'Select EEPROM' button. Then a subsequent 'Device Read' will read the 5131 into the Edit Buffer.
A subsequent Save to File will write the contents to a HEX file.

Note that you must keep an eye on the 'EEPROM Buffer Information' pane.
The 'Operations Flow' pane alters when you are in EEPROM access mode.

And after you have been using the EEPROM, remember to set back to FLASH for burning your new program. Life is probably an awful lot easier if you use BATCHISP.EXE instead of FLIP. Personally, I find that FM.EXE is a lot easier than FLASHMAGIC for the NXP chip.

Do not worry about the 5131 page modes. From the look of your application you are only going to be writing to EEPROM on an occasional basis. So you might just write one byte at a time.

If you want to use the page mode, you need to write a wreeprom_buf(char xdata * ads, char *source, char length) function. This will make sure that you write the current page before you cross into the next page. Exactly the same process as you use when writing multiple bytes to an I2C eeprom.

A Tip:
If you want to display something on an LCD, choose a Name or something else that is humanly readable. If you want to display some arbitrary binary value, use sprintf(buf, "%02X", value)

David.