“Be” is used because the verbs “request” and “suggest” activate the subjunctive mode. To activate the subjunctive mode, you could use verbs that “demand” something or for a “recommendation”. Here are some verbs: Insist, command, order, request, urge. “whether” and “if” may activate the subjunctive mode for alternative choices or uncertainty when saying something. “Whether he be right or wrong does not matter.” Hope that helps.

It sounds weird because the subjunctive mode is not commonly used in spoken English. The subjunctive mode has many uses, but one of its main uses is uncertainties. "Whether he is right or wrong." is ungrammatical. Why? It is wrong because "is" talks about the present; the present tense is used for TRUE facts (and other things). We don't know yet whether he is right or wrong (note that here I'm not making a statement, so "is" is correct), which means it is not a true fact yet, so it is an uncertainty; the subjunctive mode needs to be used. However, in an everyday conversation, it is OK/acceptable to say "is" instead of "be". For formal or academic purposes, it is wrong to say "is", though. Keep in mind that the verbs that give an order don't necessarily need an uncertainty: "I order (that) you be there." Here, it doesn't matter about the uncertainties; you MUST be there whatever happens. No excuse :) In your other question, "insisted" can either mean "demand", which is an order or "persist", which is not an order. In the context of your sentence alone, we don't really know what it is, so let's assume that it is an uncertainty so that you will know both the order and the uncertainty. "He insisted that a second red flag is added..." is wrong. Why? It is wrong because at the time that "he" insisted/persisted, "he" didn't know whether what he asked was going to be accepted or not, which means that when he asked for a second red flag to be added, he was not sure to get his request accepted, so it was/is an uncertainty; the subjunctive mode is normally needed. In formal or academic situations, it is wrong not to use the subjunctive mode here, as well. As you can see, it all depends on the context whether you can take the liberty of using the subjunctive mode or not. With friends and family, you do not really need to as you may sound like you want to "show off" your knowledge; in formal/academic situations, don't hesitate, at all! I hope that makes sense!