Hi,
1. The program/device interface with SPD EEPROMs via SDA, SCL, SA#, VDDSPD, and VSA lines which are present on all DIMM modules, so all kinds of DIMMs up to DDR4 are currently supported. Full size DIMMs and SODIMMs use the same EEPROM chips, so there's no difference for the program what module types they are used on. An adapter is needed, as DIMM and SODIMM pinouts differ.
(You link wasn't posted, but if you meant to buy something like this, then yes, this will work:
https://m.media-amazon.com/images/I/613ts61Ts9L._AC_SL1000_.jpg)
2. If the DDR4 EEPROM isn't write protected, then yes, it can be written to, but if it is, the protection can be removed with the program to enable writing. DDR4 not supporting permanent write protection is per Jedec's specification. EEPROMs used on DDR3 and prior could be reversibly or permanently write protected via a software command, but starting from DDR4 software permanent write protection was removed, so DDR4 of any type only supports reversible write protection.