When the subject of a sentence is an older person, it is usually treated as a plural noun. This is a way of showing respect. This is also why the sentence uses हैं instead of है.
राज का दादा बूढ़ा है। would be a grammatically correct sentence but it may be seen as disrespectful.
Not when used in its correct context. However, addressing someone just as 'बूढ़ा' or 'बुढ्ढा' is definitely offensive (similar to how addressing them as 'old man' might be). Also, the word बुज़ुर्ग is a more respectful way of referring to the elderly compared to बूढ़ा/बूढी.