There are a number of things wrong with your trigger.
Firstly, the error about Unknown table 'table_1' in field list refers to your use of table_1.txt_avl in the WHEN clauses of your CASE statement. It seems you want to access the value of txt_avl in the row being inserted, and to do that, you use new.txt_avl instead.
This change alone however isn't enough to get the trigger to work. If you make this change to your trigger, you will get the following error (or at least I did, anyway):
ERROR 1235 (42000): This version of MariaDB doesn't yet support 'multiple triggers with the same action time and event for one table'
The problem here is the use of an UPDATE statement that updates table_1 within an AFTER INSERT trigger on the same table. If you want to set the value of a column in the new row, you can do this by setting the new value of the column, using SET new.v1order = LC_VAR;. However, you must also change the trigger to fire BEFORE INSERT instead of AFTER INSERT.
You can't change a column like this in an AFTER trigger, as by the time your trigger has been called it is too late - the data has already hit the database.
Once this is done the trigger appears to work. Here's a quick session where I create a table and your trigger, insert some data and verify that the trigger fired and did something:
MariaDB [blah]> create table table_1 (v1order int, v1Pkey int, txt_avl int);
Query OK, 0 rows affected (0.18 sec)
MariaDB [blah]> DELIMITER $$
MariaDB [blah]> CREATE TRIGGER `FMY_NUM` BEFORE INSERT ON `table_1`
-> FOR EACH ROW BEGIN
-> DECLARE LC_VAR INTEGER;
-> CASE
-> WHEN new.txt_avl>= 7 THEN SET LC_VAR = 5;
-> WHEN new.txt_avl > 9 AND new.txt_avl < 5000 THEN SET LC_VAR = 3;
-> WHEN new.txt_avl > 11 AND new.txt_avl < 3000 THEN SET LC_VAR = 2;
-> ELSE SET LC_VAR = 1;
-> END CASE;
-> SET new.v1order = LC_VAR;
-> END;
-> $$
Query OK, 0 rows affected (0.06 sec)
MariaDB [blah]> DELIMITER ;
MariaDB [blah]> insert into table_1 (v1Pkey, txt_avl) values (1, 8);
Query OK, 1 row affected (0.03 sec)
MariaDB [blah]> select * from table_1;
+---------+--------+---------+
| v1order | v1Pkey | txt_avl |
+---------+--------+---------+
| 5 | 1 | 8 |
+---------+--------+---------+
1 row in set (0.00 sec)