When creating an OWL ontology, what are the consequences of declaring :
ex:myHasPart a owl:ObjectProperty .
ex:myHasPart owl:subPropertyOf <http://purl.org/dc/terms/hasPart> .
From the two triples you write, we can just say that if you have a triple <x> ex:myHasPart <y>, then we can infer <x> dc:hasPart <y> (in both OWL 1 and OWL 2, using either direct semantics or RDF-based semantics).
But the complete consequences of the axioms can only be fully estimated by knowing what else is in your ontology. If you only have those two triples, your ontology is not in OWL (1/2) DL. But reasoners won't care, they'll usually assume that dc:hasPart is an object property too, and things will be just fine. Also, if you have an additional statement saying:
dc:hasPart a owl:AnnotationProperty .
then the ontology is not valid OWL DL, nor OWL 2 DL, no matter what else you have in it (an IRI can only be denoting at most one type of property, annotation, datatype, or object). It is still usable in many OWL tools that will not care (like Protégé, for instance). Reasoners probably don't care either, as long as you don't use the properties in bizarre ways, e.g., with both literals and non literals.
To answer specifically to your questions:
In OWL 1 : I read from http://bloody-byte.net/rdf/dc_owl2dl/ that it implies that ex:myHasPart will be an AnnotationProperty. Can it be both an ObjectProperty and an AnnotationProperty ?
No, the two triples imply that ex:myHasPart is an owl:ObjectProperty. In an OWL 2 ontology, a property cannot be object and annotation at the same time. In an OWL 2 Full ontology, they can (an OWL 2 Full ontology is just an RDF graph). In OWL 1, an OWL DL ontology cannot have an object property that is an annotation property at the same time, while an OWL Full ontology can.
In OWL 2 : What would be the consequences/entailments ?
Pretty much the same thing as in OWL 1, but it depends what semantics you are using: Direct Semantics or RDF-based semantics? The direct semantics only applies to OWL 2 ontologies, not to OWL 2 Full ontologies, so you would have to, at least, declare your ontology <yourOnto> a owl:Ontology and declare the term dc:hasPart as an owl:objectProperty. Reasoners can be more tolerant. In any case, there is not going to be anything bad happening, unless you are messing up with the rest of the ontology.
How can I choose in Protégé to work in OWL1 or OWL2 ?
You can choose a version of Protégé that does not support OWL 2 (if you can still manage to find one). Otherwise, any version after 4.0 uses OWL 2. But from a syntactic point of view, there is no reason to restrict to OWL 1. The reasoning is done independently from the editor in Protégé, so whether you have OWL 1 or OWL 2 reasoning depends on which plugin you have. However, OWL reasoners that do not support OWL 2 are simply obsolete, there is not much sensible reasons to still use them.
Additionnally, what are the consequences in terms of OWL-DL vs. OWL-Full ? would my ontology still be at the OWL-DL level ?
I think I already explained this.
Is it a good practice to declare subProperties of dcterms like this ? I think I need a general explanation on this.
I do not see any reason why it would be considered good practice, but I don't see a reason to actively prevent people from doing so.
What could be the alternative to link ex:myHasPart with dcterms:hasPart ? rdfs:seeAlso ? skos:broadMatch ?
Your suggestions seem valid to me. But do you really need to make that link explicit?