Auto TimeStamp new entry to DB (phpMyAdmin)

That is correct. In SQL code that would be:

CREATE TABLE `table` (
    ...
    `date` TIMESTAMP NOT NULL DEFAULT CURRENT_TIMESTAMP, 
    ...
)

Yes, this method is correct:

 create table t(Id int, ts timestamp default current_timestamp)

 insert into t(Id) values (1)

 select * from t

 1;2010-09-01 09:20:09:000