What place do ontological arguments for God have in Christianity?

The greatest thinkers of the time (including the great theologian Thomas Aquinas, the Doctor Universalis of the Catholic church) rejected it. To quote from Wikipedia,

The first critic of the ontological argument was Anselm's contemporary, Gaunilo of Marmoutiers. He used the analogy of a perfect island, suggesting that the ontological could be used to prove the existence of anything... Thomas Aquinas later rejected the argument on the basis that humans cannot know God's nature. David Hume offered an empirical objection, criticising its lack of evidential reasoning and rejecting the idea that anything can exist necessarily. Immanuel Kant's critique was based on what he saw as the false premise that existence is a predicate. He argued that "existing" adds nothing (including perfection) to the essence of a being, and thus a "supremely perfect" being can be conceived not to exist. Finally, philosophers including C. D. Broad dismissed the coherence of a maximally great being, proposing that some attributes of greatness are incompatible with others, rendering "maximally great being" incoherent.

Another point is that proving the existence of a deity would still not show that deity to be the Abrahamic God. Most of the philosophical arguments for the existence of God necessarily suffer from this shortcoming (from the Christian POV that is)

At present, not much of one.

The ontological argument along its traditional lines, wherein God is allegedly proven to exist because the greatest being is an existent being, has been criticized by numerous Christian theologians. Most notably, I think, is that St. Thomas Aquinas rejected it on the grounds that only God could know God's nature; hence, any ontological argument humans could create or understand is moot.

Modern public debates between reputable Christians and atheists generally consist of restatements and clarifications of St. Thomas's proofs. Take Prof. Peter Kreeft's debate with Prof. Michael Tooley as an example.

Regarding your first question, there is a key difference between your two examples. They would be more parallel if you framed the first one like this:

God is the most perfect possible being that can be thought of. Non-existence or a lack of regular competition in barbecuing championships would be an imperfection. Therefore, God exists and regularly competes in barbecuing championships.

Neither competing in barbecuing championships nor attacks on you are essential to being a perfect being or the most scary being, respectively. However, the most scary being could be argued to be one that exists, rather than one that doesn't, so:

A monster is the most scary possible being. A being that exists is scarier than one that doesn't exist. Therefore monsters exist.

If we hold existing as absolutely essential to monsterhood over other possibly scary attributes, then this holds true. But, we may be able to think of things that seem scarier that don't exist (say a plague of Tribbles on Kronos) than those that do exist, so one might argue that existence isn't essential to monsterhood.

I think Anselm makes a stronger argument towards God, however. In his rejection of Gaunilo's critique, he notes that talking about God is different than talking about the most perfect island. At the very least, we know that the definition of perfection is tied intricately to God in the Christian understanding of God, so whatever God is must be more perfect than what God isn't.

As @justbelieve notes the real rub with the argument is that it cannot prove a particular deity, even if the argument works. We might find that the most perfect possible being -- with existence being the key attribute to that being's essence -- might be very much unlike what we want to label God in a hypothetical reality where God didn't exist.

Speaking of hypothetical realities, Alvin Plantinga has a revised Ontological Argument utilizing modal logic that deals with the idea of multiple realities. Perhaps someone who is a braver soul than I would like to explain it.

Regarding your second question, the Ontological Argument's major proponents have included St. Anselm, René Descartes, Karl Barth and Alvin Platinga, just to name a few. Contra the argument within Christian tradition would be a great many others, including St. Thomas Aquinas (cf. Summa Theological 1.2, especially questions 1 for the rebuttal and 3 for his alternative).