You do indeed multiply the smite damage and the strength modifier.
Sometimes you multiply damage by some factor, such as on a critical hit. Roll the damage (with all modifiers) multiple times and total the results.
Note: When you multiply damage more than once, each multiplier works off the original, unmultiplied damage. So if you are asked to double the damage twice, the end result is three times the normal damage.
Exception: Extra damage dice over and above a weapon’s normal damage are never multiplied."
Using the other example: your greatsword rolls 2d6 on attack and 1d6 fire damage, you crit and roll 8 on the 2d6 and 3 on the fire damage, you also have a +4 str bonus. So you add the 8 and your strength bonus which adds 6 damage since you are wielding a two handed weapon getting 14. We will also assume that you have used Smite and your target was Evil and your paladin level is 5 since this was your question. We add the 5 to the 14 and get 19. Since you crit we do it again. Rolling the 2d6 again we get 7. Add our 6 for strength and 5 for smite for 18. Add the two damage rolls together for 37 and then add the 3 fire damage, this makes for a total of (8+6+5)+(7+6+5)+3 = 40