Paso por la tienda después del trabajo.

Paso is present tense (yo paso = I pass by / I stop by), which is often used for routines or near-future plans depending on context.
If you mean a completed action in the past, you’d use preterite: Pasé por la tienda después del trabajo (I stopped by the store after work [that day]).

Usually no. Spanish commonly drops subject pronouns because the verb ending already shows the subject: paso = I.
You might add yo for emphasis or contrast: Yo paso por la tienda, pero él no (I stop by the store, but he doesn’t).

Pasar por literally means to pass by/through, but in everyday speech it often means to drop by/stop by briefly.
So Paso por la tienda commonly implies a quick stop rather than a long shopping trip (though context can change that).

Por emphasizes passing by/along the way or making a stop “on your route.”
If you want to say your destination is the store, Spanish often uses ir a: Voy a la tienda después del trabajo (I’m going to the store after work).
With pasar por, the store is more like a stop during your movement.

La tienda suggests a specific store known in context (your usual store, the store you both have in mind).
Una tienda would mean a (some) store, not a specific one: Paso por una tienda (I stop by a store [some store]).

In Latin American Spanish, tienda commonly means store/shop. In some places it can also refer to a small neighborhood shop (like a convenience store).
(Separate meaning: tienda de campaña = tent, but tienda by itself is generally store.)

Yes. With present tense, Paso por la tienda después del trabajo can describe a habit (I stop by the store after work) or a typical plan.
If you want to make it clearly “today,” you can add hoy: Hoy paso por la tienda después del trabajo.

• después = afterwards/then (no noun directly after it)
  
  • Después paso por la tienda (Afterwards I stop by the store)
• después de = after
  • noun/phrase
    
    • después del trabajo (after work)