Literally, de esta mañana means of this morning.
In natural English, that often becomes:
So Las moras de esta mañana is basically the blackberries from this morning or this morning’s blackberries.
This is a very common Spanish way to connect a noun with a time expression:
Because mañana is a feminine noun when it means morning.
So the demonstrative has to agree with it:
Even though mañana ends in -a, that alone does not always guarantee gender in Spanish, but in this case it really is feminine: la mañana.
Both estar and ser can translate as to be, but they are used differently.
Here, estaban muy dulces describes the state/condition of the blackberries at that moment: they were very sweet. That is why estar is the natural choice.
So:
For food, Spanish very often uses estar with adjectives like bueno, rico, dulce, salado, etc. when talking about how it is on that occasion.
Estaban is the imperfect, while estuvieron is the preterite.
In this sentence, the imperfect is used because it describes a background quality or ongoing state in the past: the blackberries were very sweet.
With descriptions in the past, Spanish very often prefers the imperfect:
So estaban fits the descriptive feel of the sentence.
Yes, you could, but it means something slightly different.
The first version attaches this morning directly to the blackberries.
The second version attaches this morning more to the whole situation.
In many contexts the practical meaning is very similar, but grammatically the focus is a little different.
Yes, very common.
Spanish often uses de + time expression where English might use a possessive-style structure:
So de esta mañana is completely natural and idiomatic.
Yes. A literal translation helps you understand the structure, but natural English may be different.
Depending on context, natural English could be:
The Spanish structure does not always map word-for-word onto English, so it is normal to understand de esta mañana as a time label attached to the noun.
Because of number agreement.
Las moras is third person plural, so the verb must also be plural:
So estaban clearly matches las moras.
The sentence itself is standard Spanish and would be understood widely, not only in Spain.
That said, a learner of Spanish from Spain might notice that fruit vocabulary can vary by region. In Spain, mora is a normal word, and the overall structure of the sentence is very typical of everyday Peninsular Spanish:
So even if the sentence is not uniquely Spanish, it fits very naturally with Spanish as used in Spain.