1) Can I use my EAV-xxxxxxxx Username/Password in ESET smart security?
You can use an ESET Smart Security license to download and update ESET NOD32 Antivirus but not vice-versa.
2) Are the definitions (update files) are the same as ESET?
Yes, update files are same, however, not all product versions may necessarily have the same version of modules installed at all times
3) Do all the components of ESET get updates like firewall or... ?
Only components used by a particular product are updated, ie. ESET NOD32 Antivirus will not download firewall module updates.
4) How can I check if it's OK or not? Is there any way to make sure? Please tell me about it.
If an update attempt fails, you'll receive a message about this. You can also compare the version of the installed virus signature database with the version listed on ESET's website if you want to make sure you're fully updated.