Is reading books and listening to the explanations of Sheikh Al-Uthaymeen via recordings a correct method for seeking knowledge, and is it valid to consider the Sheikh as "my Sheikh" without direct meeting or attending his gatherings?

Islamic knowledge is built upon three pillars: memorization, comprehension, and analysis with inference. Reading, attending lessons, and listening to lectures are means to acquire these pillars. The best of these is reading, because it combines the other means. Reading must be a means to develop the three pillars of knowledge, not an end in itself. Similarly, listening to recorded lectures is beneficial for compensating for the inability to meet scholars, and for saving time and effort, and it has many benefits in seeking knowledge. As for using the term "my shaykh" for a scholar whom you have not continuously accompanied, it may be blameworthy if it creates the illusion of actual companionship, unless the listeners understand that what is meant is discipleship through listening to lectures and tapes, in which case there is no harm.