I'm a little curious why you think you need a branch for a corn snake. If it's for decoration, go for it!, but corn snakes are not arboreal and don't need to live with tree branches. In the snake world, corn snakes are one of the most low-maintenance ones out there.
You can stretch out the use of aspen by spot cleaning. You never reuse chips once they've gotten dirty, but you can scoop out the poop along with the surrounding chips and leave the rest in there for a while. It depends on how messy your snake is and how often it poops. One of the big reasons fresh water is a must is because snakes will often go to the bathroom in their water dish.
And as far as sexing goes, visually sexing snakes is not reliable at all. You should take them to a vet or experienced keeper to have the popped or probed. Popping or probing is not something you want to try yourself because you could hurt the animal if you don't know what you're doing.
Last, egg laying, it is rare for a female who has not mated to lay eggs. It's not unheard of of, but it's not the typical behavior.
^ I definitely agree with this.
There is some truth about females having shorter tails, but it's not noticeable to the eye unless you've looked at a lot
of tails or sit down and count scales. Can you tell from my pictures which is the male and which is the female?
Popping is reserved for very young snakes; if you try to pop on your own on a non-hatchling, you risk damaging its bones.
And to expand on tromboneplaya's last part--all sexually mature females will make eggs. If the eggs are unfertilized, the snake will re-absorb them instead of laying them, but there is a small chance she'll lay the dud eggs.