Do mobs get stronger the further your away from your spawn point?

To my knowledge no. Mobs themselves are not given any additional powers and I do not believe it becomes more likely for them to be spawned with better gear or for certain mobs to spawn more frequently.

The Minecraft wiki has all the information about mob spawning you could want:

https://minecraft.gamepedia.com/Spawn#Spawn_Cycle

There is a "regional difficulty" mechanic such that the longer players spend in chunks, the higher the regional difficulty is, although there is a cap to this difficulty and it doesn't dramatically affect your experience vs world difficulty.

https://minecraft.gamepedia.com/Difficulty#Regional_difficulty

This would imply the opposite of what you're suggesting. Rather than traveling increasing difficulty, staying in place is what increases it.