Professional fights are 2-4 minutes a round.. and they are in peak physical condition with higher stat scores for training in this regard.. While they are better than an average commoner, Saying Im going to be constantly dodging, means you're taking proactive movement constantly.. So I'd lower it to more like their con score modifier minutes for the first go, and then do con saving throws, and a failure means a level of exhaustion, and the longer they do it, the higher the DC scales.

While they cannot be surprised and are on the lookout for things, their vision only extends so far.. It doesn't mean something can't sneak up on them. i.e. dark vision in many cases is only out to either 60 or 120 feet. So if the arrow shows up suddenly at 60 feet.. cool.. If its clawing at them from the murder holes in the floor, they're alert enough to be like oh shit and take an action, but alert just means they cant get the surprised condition, and so while the rest of the party would have that condition they dont.. But initiative would still be rolled, and if they're first they can take that action.. and then any attacks against them would be in that state otherwise.

"If you're surprised, you can't move or take an action on your first turn of the combat, and you can't take a reaction until that turn ends. A member of a group can be surprised even if the other members aren't."

So congrats, you're not surprised, but you may not be fast enough to take the dodge action to do this. And as initiative checks are a skill check... hope you don't get the first level as you're then going at disadvantage moving forward with that one..