For example, would a level one geodude with sturdy faint from a level 100 blissey using seismic toss?
Yes, sturdy will activate and save your Pokemon as long as it was at full HP and it took damage from an actual move. However, it won't activate for any other type of damage, like weather, status conditons (though neither are going to KO you from full unless you're Shedinja) SOURCE
Example Battle:
