Can SQL Server avoid subqueries or joins if other where clauses return enough results?
Does query optimizer manage to prioritize the first clause, avoiding
  to do the subquery if the simpler clause is true for at least 1 record.


It can do 

create table customers(id int identity primary key, firstname varchar(10)) ;
insert into customers values ('John'),  ('Johnathan'), ('George');
create table customer_emails(id_customer int, email varchar(100));
insert into customer_emails values (1, 'John@example.com'),  (2, 'Johnathan@example.com');


In the plan I got the first row matched the LIKE 'John%' predicate and the scan against customer_emails was not executed at all.



However your question is phrased as 


  if the simpler clause is true for at least 1 record


That would imply that the simpler WHERE clause is evaluated in its entirety and only if that fails is the second one evaluated.

For 

SELECT TOP 1 *
FROM customers
WHERE firstname LIKE 'George%'
   OR id IN (SELECT id_customer 
             FROM customer_emails 
             WHERE email like 'George.doe@%' );


Three rows were processed before one was found matching LIKE 'George%'and there were two ensuing scans on customer_emails

Rewriting as follows...

create index ix on customers(firstname) include(id)

SELECT TOP 1 *
FROM customers with(forceseek)
WHERE firstname LIKE 'George%'
   OR  firstname NOT LIKE 'George%' AND  id IN (SELECT id_customer 
             FROM customer_emails 
             WHERE email like 'George.doe@%' );


...  happens to give a plan where the operators in the shaded area for the IN part don't get executed if the simpler predicate returns a row but this plan isn't guaranteed.