First AG batch in the fermentor - now a couple questions

I brewed an AG Amber Ale using the recipe in Brewing Classic styles last Saturday. It went very well and is merrily burping out the blow off tube as we speak. If the beer comes out as good as I hope, I intend to enter it into the upcoming National Homebrewers Competition. I don't imagine I have much of a chance but I am really interested in the feedback I will get on my beer.

I got a 4 cu. ft. fridge for Christmas from my son & SWMBO. My carboy will fit nicely in there so I am thinking of cold crashing this batch which would also free up my carboy so I can use it on a Belgian recipe.

So here's the questions;
Normally I let my beer sit in primary for three weeks then bottle. If I cut that a few days short to cold crash, do you think it would matter? I think I want a very clear beer for a competition - will cold crashing give me that?

If I do shorten the primary a few days and bottle weekend after next, I can brew the Belgian that same weekend. That would let it ferment for three weeks, get bottled and then be three weeks old for the first round of judging. Would that be pushing it (age wise?)

Like I said at the beginning, I'm not expecting much more than good feedback from the judges, but if I'm short cutting things, I could just slow down a bit and submit just the Amber.

What do you think?

Cold crashing can definitely cause chill haze, regardless of whether you used Irish moss or not. If you are going for a clear beer, I would highly suggest a secondary stage.
As for making time for a Belgian, those kinda beers need time to age. If you are using a true Belgian strain and don't want it possibly bringing down your first batch of pride, age the Belgian for a few months and see how it goes. If you like it, enter it next year and plan further in advance.

I guess I'm confused, I thought cold crashing would help reduce chill haze. Why would it induce it???

The main problem I see with cold crashing is knocking out your yeast before bottle conditioning. If you plan to keg then you don't have to worry about that. I have not used the approach you're talking about but I tried a beer at a competition and it was super clear. The brewer said he cold crashed at the end of primary, racked to secondary and dry-hopped then cold crashed it again.