Can someone who's gained access to my iCloud account prevent me from receiving text messages? Can someone use my iCloud account to prevent me from receiving messages on my devices from specific people?

I can get calls from a particular person but no text messages for a few days now.
No.

iCloud has nothing to do with Text Messages (SMS).

iMessages are sent to all your devices, so even if someone was using your iCloud credentials, you would still receive the iMessages.