Yes, it can be (and is accepted). Often when in English the present continuous is used, the simple present is used in Dutch.
There is a "continuous" in Dutch "zijn + aan het + infinitive" though that generally is used to emphasise that it is happening at this moment.
The meaning of de and het is in both cases the. I suggest you have a read through this topic: https://www.duolingo.com/comment/3732938 It gives an explanation on when to use de or het. It is up to you to decide if you make use of those rules or if you think it is easier to just learn what word uses de and what word uses het.
In short (which always applies)
- Plurals always use de
- Diminutives (singular) always use het
- Plural diminutives are still plural and hence use de
Does Dutch lack declensions entirely, or is "de appel" coincidentally identical in the nominative and accusative cases?
No it is not supposed to be silent. We really distinctively pronounce each word in the sentence (dutch native speaker, but just going through the course for fun:p)
It's because 'de' is pronounced /də/ with a schwa sound like the 'a' in about. Because it is unstressed and because the 'a' in appel is stressed it can be difficult to hear, sounding more like « d'appel ».