How to return some * data with time interval?
Maybe a combination of the functions SECOND() and MOD() will help.  Using your test data  as follows (dbfiddle here, also tested with MySQL 5.7):

create table t_ (
  id_ int primary key
, date_ datetime
);

insert into t_ values
(1,'2017-07-10 10:11:10'),(2,'2017-07-10 10:11:15'),
(3,'2017-07-10 10:11:20'),(4,'2017-07-10 10:11:25'),
(5,'2017-07-10 10:11:30'),(6,'2017-07-10 10:11:35'),
(7,'2017-07-10 10:11:40');


We can get the "second" value of the date_, for every 15th second, like so:

select second(date_) 
from t_
where mod( second( date_ ), 15) = 0 ;

-- output
+---------------+
| second(date_) |
+---------------+
|            15 |
|            30 |
+---------------+


The SECOND() value of the "last registered date" would be ...

select second( max( date_) ) from t_ ;

-- output
+-----------------------+
| second( max( date_) ) |
+-----------------------+
|                    40 |
+-----------------------+


Take the "last registered date" into account:

select *
from t_
where mod ( 
  ( select 60 - second( max(date_) ) from t_ ) 
  + second( date_ )
  , 15
) = 0 ;

-- (your required) output
+-----+---------------------+
| id_ | date_               |
+-----+---------------------+
|   1 | 2017-07-10 10:11:10 |
|   4 | 2017-07-10 10:11:25 |
|   7 | 2017-07-10 10:11:40 |
+-----+---------------------+


This query should also work if there are "gaps" in the date_ (the MOD() function does not care) eg if we DELETE everything from the table t_ and insert the following rows:

-- IDs 5 (10:11:20) and 11 (10:11:50) missing
insert into t_ values
(1,'2017-07-10 10:11:00'), (2,'2017-07-10 10:11:05'),
(3,'2017-07-10 10:11:10'), (4,'2017-07-10 10:11:15'),
                           (6,'2017-07-10 10:11:25'),
(7,'2017-07-10 10:11:30'), (8,'2017-07-10 10:11:35'),
(9,'2017-07-10 10:11:40'), (10,'2017-07-10 10:11:45'),
                           (12,'2017-07-10 10:11:55'),
(13,'2017-07-10 10:12:00'),(14,'2017-07-10 10:12:05');


Our query will return:

mysql&gt; select *
    -&gt; from t_
    -&gt; where mod ( 
    -&gt;   ( select 60 - second( max(date_) ) from t_ ) 
    -&gt;   + second( date_ )
    -&gt;   , 15
    -&gt; ) = 0 ;
+-----+---------------------+
| id_ | date_               |
+-----+---------------------+
|   2 | 2017-07-10 10:11:05 |
|   8 | 2017-07-10 10:11:35 |
|  14 | 2017-07-10 10:12:05 |
+-----+---------------------+


If there are (date/time) duplicates, then ... you will see them in the output.  You can add a BETWEEN clause (as @Jess suggested) at the end of the query, if you need a "master interval".

UPDATE
It seems that you also want to capture values that are smaller/greater than the (exact multiples of) 15 seconds - you have mentioned 18,32, and 57 (seconds), respectively.  With a slightly modified query, you can pick up these values, too.  Eg the following query has a "tolerance" of +/- 3 seconds eg if the "exact" second is 15, we'd also get 12,13,14,16,17,18.  Notice that this "frame" moves according to the MAX() value ie your "last registered" time.

select *
from t_
where mod ( 
  ( select 60 - second( max(date_) ) from t_ ) 
  + second( date_ )
  , 15
) in (0,1,2,3,12,13,14) ;


2 sets of test data (dbfiddle):
"last registered" near the full minute , 
"last registered" at the 30 second mark