Yes, without an IMCr or IR you can transit class D VFR or SVFR.
With either you can request an IFR crossing.
You may be given an IFR reporting point but in practice that is less likely and a nuisance. For example if I want to transit Solent at say FL50 I will simply tell them that I am at FL50 wishing to transit on current heading but can accept vectors if required. More often that not the transit will be cleared on current heading or a differnt heading will be given and / or a different altitude. That causes the least disruption to my flight and more than likely aircraft that are arriving and departing can be readily vectored around me. Moreover if you are high enough the chances are arriving and departing CAT will be well below your level anyway so the only traffic to efect will be other en route traffic.
Without being pedantic do you mean "clarrifying you are VFR". You are either operating under VFR or IFR. Obviously to be under VFR the met conditions must also be VMC but to be IFR the met conditions could be either VMC or IMC. Therefore if you ask for a VFR clearance the controller will know you are in VMC, however if you ask for an IFR transit he doesnt know (and doesnt really care). If you like you can request an IFR clearance and tell the controller you are in VMC but 1) it will have no effect on how he handles your transit and 2) he is entitled to assume you are happy entering IMC so if you have indicated you can accept vectors he may either vector you into cloud or put you into cloud by asking you to adjust your level. If on the other hand you request a VFR transit he will usually say in his clearance if you cannot maintain VMC then say so and he will sort you out.