If pepsin breaks down proteins smaller, then how do stomache cells which are proteins not get destroyed?

Maybe you should find out what cell walls are actually made of  [clue:  it's not protein]

Pepsin is not released in its active form.  It is released as a precursor called pepsinogen, which only becomes an active protease when in the presence of stomach acid (it's the low pH that does it).

The epithelial cells have a gastric mucosal protective barrier.  The barrier has two elements:
1 - Mucin the mucus glycoprotein, secreted by gastric mucosal cells to bind MBP and block proteases. MBP without mucin is susceptible to the proteolytic degradation with pepsin.

2 - The mucin binding protein (MBP) is part of the apical epithelial membrane and it holds the mucin to protect the lining cells.

Yes, both work to prevent autodigestion of the epithelial lining of the stomach. Pepsin is released by gastric chief cells as pepsinogen an inactive form which is then activated in the presence of HCl. There are numerous mucus-secreting cells in the stomach as well called surface mucous cells and mucous neck cells. These secrete a protective mucus that lines the stomach surface and protect it from acidic and enzymatic digestion.
Some drugs (NSAIDs) can interfere with mucus production which then removes this protective lining and can lead to ulcers (through acid and enzyme damage).
By the way, animal cells don't have cell walls - they have cell membranes (or plasma membranes) and while they are largely composed of phospholipids, there are also proteins present.