"①plan" is correct in this case. A plan is a list of steps determined in advance in order to achieve a goal: the entire list of things Tony has to do, and possibly the sequence in which he does them make up the plan. On the other hand, if you are talking in general about somebody without going into detail, the colloquial way to express that is "he has plans for the day".
I agree that ② should be "will be", unless he is busy now as well, or usually busy.
The email question is tough, because "email" stands for the entire system of electronic communication, such as in "Communication by email is almost instantaneous.", as well as for each individual message: "I sent an email to my boss about our meeting". And sometimes it's also used as a generic term for the software you use to read, write, send and receive email messages.
I prefer to say "check email" if I am looking at my entire inbox because it feels more like a mass action I am performing on my inbox, instead of something I do with individual email messages -- I might in fact not do anything with a lot of the incoming messages, but maybe I am only looking at work-related items, or i am just waiting for one single email from a friend. But I see a lot of "check emails" from native speakers as well. As with anything new, the terminology may well be in flux.
Btw, "he is going to receive for his English exam" is not an English expression, and I don't know what you mean by that.