In the first question, the word does is the verb and does has the 3rd person singular Present Tense s. So the word have can be regular, without the s. In the second question, the word has IS the verb so it needs the third person singular Present Tense s, but the word have/has is irregular and has two forms, you have to use the s form, so use the word has.

To build off of what Su.Ki. said: The difference in the inflection of "to have" in these sentences has to do with which verb the auxiliary verb is commanding in these sentences. When "do" acts as an auxiliary verb, the verb it commands changes to its base form regardless of whether the subject is 1st/2nd or 3rd person. "He does have a dog." vs "He has a dog.", "He does eat." vs "He eats." In the first example, the auxiliary verb here is commanding the verb "to have" The best way to figure this out would be to reverse the subject-auxiliary inversion and convert the sentence into a statement. "He does have a dog or cat." In the second sentence, you have 2 clauses. Clause 1: Do you know Clause 2: If Raymond has a dog or cat If we again reverse the subject-auxiliary inversion, we have "You do know" as the first clause which shows that the auxiliary verb here is commanding the verb "know" Here is an example of the second clause containing the auxiliary verb "do": "Did you know that Raymond does have a dog or a cat?" Here you can see the verb has reverted to its base form due to being commanded by the auxiliary verb "do" here. Hope this helps clear things up and remember to always separate clauses and track auxiliary verb+verb relations :)

Both of these questions are in the standard present simple interrogative form : do/does + subject + verb in its base form (without 's') Does Raymond have......? Do you know ...........? In the first sentence, Does Raymond have......? is the question, so 'have' is in the base form because this is part of the interrogative structure. In the second sentence, 'Do you know..' is the question. The phrase which follows this, '(if) Raymond has a dog or a cat', is NOT in the interrogative form. 'Raymond has a dog or a cat' is in the standard affirmative statement form (subject + verb), so the verb has to agree with the subject.