QDoes "you may send 1 blue cheer from your archive to your 〈Fuwawa Abyssgard〉." of hBP03-109〈Ruffians〉 mean I can send it to hBP03-050〈FUWAMOCO〉 too?

AYes, you can.
Even though hBP03-004〈FUWAMOCO〉does not have the card names specified, as it has the extra ability "This holomem is also regarded as 〈Fuwawa Abyssgard〉〈Mococo Abyssgard〉", it allows the card to be regarded as having the specified card name, and that condition will be fulfilled.

QCan I add hBP03-050〈FUWAMOCO〉 to hand using hBP03-084〈Gorgeous PC〉 when my Oshi holomem is hBP03-004〈FUWAMOCO〉?

AYes, you can.
As hBP03-050〈FUWAMOCO〉 is both red and blue, it fulfills the condition.

QCan I use the Arts "Wanna eat together?" of hBP03-043〈Fuwawa Abyssgard〉 to reattach cheer to my hBP03-050〈FUWAMOCO〉?

AYes, you can.
Even though hBP03-004〈FUWAMOCO〉does not have the card names specified, as it has the extra ability "This holomem is also regarded as 〈Fuwawa Abyssgard〉〈Mococo Abyssgard〉", it allows the card to be regarded as having the specified card name, and that condition will be fulfilled.

QFor the Arts "The Fluffy One" of hBP03-039〈Mococo Abyssgard〉, does it get "+30" even if my center holomem is BP03-050〈FUWAMOCO〉?

AYes, it does.
Even though hBP03-004〈FUWAMOCO〉does not have the card names specified, as it has the extra ability "This holomem is also regarded as 〈Fuwawa Abyssgard〉〈Mococo Abyssgard〉", it allows the card to be regarded as having the specified card name, and that condition will be fulfilled.

QFor the bloom effect "Fuwa Floating with the Current" of hBP03-042〈Fuwawa Abyssgard〉, can I deal "20" special damage even if my only other holomem on stage is hBP03-050〈FUWAMOCO〉?

AYes, you can deal.
Even though hBP03-004〈FUWAMOCO〉does not have the card names specified, as it has the extra ability "This holomem is also regarded as 〈Fuwawa Abyssgard〉〈Mococo Abyssgard〉", it allows the card to be regarded as having the specified card name, and that condition will be fulfilled.

QFor the gift "The Rowdy Demonic Guard Dog" of hBP03-039〈Mococo Abyssgard〉, can I move my center holomem to the back position without resting it even if it is hBP03-050〈FUWAMOCO〉?

AYes, you can move without resting it.
Even though hBP03-004〈FUWAMOCO〉does not have the card names specified, as it has the extra ability "This holomem is also regarded as 〈Fuwawa Abyssgard〉〈Mococo Abyssgard〉", it allows the card to be regarded as having the specified card name, and that condition will be fulfilled.

QCan I use the ability of the bloom effect "It's playtime!" of hBP03-038〈Mococo Abyssgard〉 to add hBP03-050〈FUWAMOCO〉 from my deck to hand?

AYes, you can.
Even though hBP03-004〈FUWAMOCO〉does not have the card names specified, as it has the extra ability "This holomem is also regarded as 〈Fuwawa Abyssgard〉〈Mococo Abyssgard〉", it allows the card to be regarded as having the specified card name, and that condition will be fulfilled.

QCan I send cheer to my hBP03-050〈FUWAMOCO〉 with the Oshi skill "Moco-chan!" of hBP03-004〈FUWAMOCO〉?

AYes, you can.
Even though hBP03-004〈FUWAMOCO〉does not have the card names specified, as it has the extra ability "This holomem is also regarded as 〈Fuwawa Abyssgard〉〈Mococo Abyssgard〉", it allows the card to be regarded as having the specified card name, and that condition will be fulfilled.