What is the "ease" of the "dharma gate of ease and joy"? Can an unelightened person achieve it outside seated meditation?
Ought I imagine everything is difficult, or everything is easy? How can we be effortful in our everyday lay tasks?
Dogen means that when you practise zazen you let go of troublesome thoughts. When you do this your state is one of ease (in the sense of 'being eased' rather than 'easy').
I think Dogen would have agreed with Hui Neng on the subject of enlightened and unenlightened people : One foolish notion is enough to shut off Prajna, while one wise thought will bring it forth again.
When not doing zazen, we should still be practising awareness (some sanghas refer to work in the temple as 'working meditation'). Although sometimes we need to think about difficult stuff. Zazen is just a time when we don't have to and can let go of everything.
What does “ease” mean in the phrase “dharma gate of ease and joy”?
It means that ones five spiritual faculties are balanced.
In meditation practice, e.g. Samatha meditation it means that if the mind strays from the primary meditation object, i.e. the breath, then one does not force attention back onto the breath. Instead it comes naturally and without striving.
Joy related to practice, arises when ones practice gains momentum meaning that one spends little but right effort into cultivating wholesome habits and purifying the mind. Joy arises easily in beings that are non-attached. There is a lightness of mind because they do not carry any bagage from one moment to the next.
Ease and joy arises when one interacts with reality instead of reacting. One is seeing reality for what it is, thereby gaining freedom from mental proliferation.
The short answer to what is ease is sukha. The more unattached you are, the more effluent-free, and unbinding you are, you get to experience this positive aspect sukha.
Sukha can be translated as pleasure, happiness, bliss, or ease. It is a pleasurable experience to have. In DN 9: Potthapada Sutta Buddha says how if one could outgrow ones gross self, and enter the mind-made – the sense of self that can be developed in meditation - your defiling mental qualities will be abandoned, bright mental qualities will grow, and you will enter into this state of ‘Ease’. Then there is there is joy, rapture, serenity, mindfulness, alertness, and a pleasant/happy abiding.
Even outside of meditation you can experience such a state if you are prepared to discipline yourself as per the scriptures. You can get an idea about what it takes to discipline self in the Ganhaka Moggallāna Sutta in Majjima Nikāya.
First Buddha will advice one to come and be virtuous first – to discipline yourself from virtue, to make your life pattern good. Then he will get you to protect your six faculties (eyes, ear, etc.). For eq. when you see a figure from your eyes, low defilements such as craving and anger could be formed in you if your eyes were not disciplined. Protecting the last faculty, the mind, is to safeguard your mind without letting defilements to arise – to discipline your mind well.
The next level of discipline is when it comes to food. This is how you go about it: Now you have to eat mindfully (with the realization of true meaning of eating), inquiring wisely. That is, I am taking this food not for the strength, not for the satisfaction, not for the good looks, or not to play. I am taking this food only to continue this life, to get rid of tiredness, and as to a help to continue the solitary life away from lay life. I will free from old pains and stop making new pains by taking this food. I will continue this life and have a correct and painless life by taking this food.
At this stage you are ready for mediation, and to experience true Ease.