There seems to be a lot of confusion regarding the use of haber impersonally here, so I'll try to clean up some of it with an explanation starting from the beginning.
Haber is an auxiliary verb that means to have. It is used to form the perfect tenses along with a past participle. In this use, it is conjugated to match the subject of the sentence. It is irregular in the present (he, has, ha, hemos, habéis, han), the preterite (it is a u-group irregular with the stem hub-), and the conditional and future (it has the stem habr-).
It also has a secondary and largely unconnected usage as an impersonal verb which means there be, which in English would usually be conjugated as there is or there are. Its behavior in this usage is different from its English counterpart, so it is confusing. I don't want to overstep my bounds by trying to claim I understand why the Spanish language has evolved into having this feature, but I have learned and remembered it by using this interpretation: The noun in a sentence with the impersonal haber is not the subject, so it does not force the conjugation of the verb. The verb is always in the third person singular form (which is not ha, as in the auxiliary use, but rather hay in the present tense) because it is representing the world or existence as a singular whole.
Compare this to the English sentence "It rains in the spring and summer." The subject is not "the spring and summer", and if it were, the verb to rain would be conjugated as rain rather than rains because the subject would be plural (Notice "The spring and summer are...", not "The spring and summer is..."). Instead, the subject is it. This is known as a dummy subject because it doesn't really represent anything. What is it? The world is about the closest answer you can come up with, because it represents the state of being of things in a general sense, not any object in particular. This is analogous to how haber functions. If you say "Hay nieve en la montaña," you're saying "There is snow on the mountain," or perhaps more literally, "It has snow on the mountain." This isn't how the English sentence would be constructed, but it's similar to how you would say "It is snowing on the mountain" if the action were happening at that moment. Hence, the singular form hay is used even if there is nothing else singular about the sentence.
duoLingo is OK with "lots of kids": maybe it's a stylistic collocation issue? i.e. "kids" is more colloquial than "children," and "lots of" is more colloquial than "many"--hence, "lots of" collocates with "kids," and "many" collocates with "children." But as a point of grammar, of course "lots of children" should be accepted.
Not verbs, when "haber"is used with a past participle (http://spanish.about.com/od/verbs/a/intro_past_part.htm) then "haber" is conjugated to whatever subject and tense is needed. http://spanish.about.com/od/verbtenses/a/haber_perfect.htm
The rest of what you said is correct though. When it is used alone with a noun to mean "there" + "to be" (e.g. There is/are/was/were/will be ...) then it stays in the 3rd person singular (hay, había, habrá, hubo, hubiera, haya). http://spanish.about.com/cs/verbs/a/haber_as_there.htm
TilEulenspiegel: It would be relevant to know if you are talking about ONE thing or MORE than one thing. so................"habrá un médico en la oficina mañana" would mean "there will be a doctor in the office tomorrow"............"habrá dos médicos en la oficina mañana" would mean "there will be two doctors in the office tomorrow"..............but in either case you can use "habrá"..............note: many native speakers use this incorrectly and use "habrán", but it is not grammatically correct.
I think the use of the singular here is because it's, "There will be . . ." not "They will be . . ." It's referring to a future point in time, not the children themselves. So, if I wanted to say, "Many children will be at my house," I could say, "Muchos niños habrán en mi casa," or, "Habrán muchos niños en mi casa," but to say, "There will be . . ." one has to use the singular.
No, actually, "haber" is an interesting verb, it stays singular no matter what words surround it, it only changes in tense if needed and it only matches the noun when it's being used as an auxiliary verb (he, has, ha, hemos, han) in compound verbs aka the "perfect" tenses (e.g. He hecho=I have made) http://spanish.about.com/cs/verbs/a/haber_as_there.htm
Hmm . . . thanks for the link. According to the article, it depends on where you are, but recommends sticking to the singular form unless you hear the locals using the plural. It doesn't actually appear to be a fixed rule, but has more to do with regional dialects.
"Haber can also be used in the same way for other tenses. The general rule for the other tenses is that the singular form is used for both singular and plural objects, although it is quite common, particularly in parts of Latin America, to use the plural forms with use of plural objects. Había muchas personas en la iglesia, or habían muchas personas en la iglesia, there were many people in the church. (In some areas, habían is considered substandard, so avoid it unless you hear native speakers using it.)"
Upon further reading, it appears that the plural forms can be found in old literature -- they exist, but aren't commonly used anymore. I suppose to use them, one would sound a bit like an English speaker using Olde English.
Yeah.. haber is an interesting verb, it can mean "there" + "to be" if it is by itself: hay=there is/are, había/hubo=there was/were, habrá=there will be, habría=there would be, haya=there is (subjuctive), hubiera=there was/were (subjunctive). But when it's used in a compound verb it is translated to "have"
From my understanding the only (in the present tense) occasion that HABER is not used as an auxiliary verb is when it morphs into hay from HA (3rd person singular). This verb form, hay, can be used to express singular and plural expressions. Hay un gato aqui. Hay los gatos aqui. There is a cat here. There are cats here.
Plenty implies that there are enough for some purpose. For example, "there will be a lot of children, but we might still need more." makes perfect sense, but "there will be plenty of children, but we might still need more." is a very unnatural construction, if not all together nonsensical.
What do you mean? In a sentence like this where haber is used as an impersonal verb for existence, the only possible form is the third person singular. Those forms are only used for the compound tenses, and the fact that there are six forms isn't any different than any other tense or verb.
By the way, the yo form is habré, not hubré.
EDIT: Not as relevant anymore.
It's close, but "several" is "varios" in Spanish.