Lets see if I can help at all.
I am using a 20 gallon bucket for primary fermenation. I've been peaking into it once every couple of days because the curiosity is killing me. I've been careful not to breath into the bucket, clean hands, etc. Is this problematic?
A 20 mgallon bucket has a lot of headspace. That can become a problem when fermentation subsides. It will let lots of air in and you will lose the CO2 protective blanket. Acetobacteria infection is probably the biggest concern (vinegar).
Also,instead of using a carboy with an airlock (which I do own), the employee at my local beer supply store told me to just put a lid on the bucket, but DO NOT seal it tight. This is what i have done. Is this okay to do?
It is OK for the start, but once fermentation is over, air can then get in. It is at this stage you want it sealed. I use an airlock on everything.
Aside from a bit a semi-violent splashing of the wort into the 20 gallon bucket, I did not aerate. Is this sufficient to get the yeast going?
Lets hope it is enough. Too late now. Next time you need to shake it as much as you can, and when you think you a have done as much as you can, repeat it. While it is possible to have too much dissolved O2, you can't get that much from shaking alone.
It took about 2 days for the krausen to form, but it wasn't to boisterous (a few inches in some spots, at best). It fell only 2 days after that. Today is days 7 in the fermenting bucket and all krausen is gone. One would assume that fermentation is complete, but 2 things lead me to think otherwise:
A few inches is a lot. Sounds like it went fine.
1. There is only about 1cm of sediment on the bottom of the bucket.
Sediment forms as yeast and trub drop out. 1 cm is not out of the ordinary. I usually have about 4 pints of trub in 7 gallon batches.
2. The temp fell drastically in the past 24 hours. I had the window open in the fermenting room and today I checked the temp in the bucket and it was a frightening 8c!!!!! The room is now warming. I closed the window, turned the heat on.
That will stall the yeast and make it drop out. Do NOT rack the beer to secondary until you get it back to mid 60s F temps. The cold probably stalled the ferment. Let it warm back up, maybe gently swirl the fermenter (when warmed back up).
Can you give me any advice?
Did the drop in temperature kill the fermentation process?
No, it just stalled it. The yeast will wake up when they get warmer.
Is there anything I can do to rouse the yeast?
Swirl the fermenter GENTLY when it has warmed up.
Is 1 cm of sediment normal?
Seems fine to me.
Is the bucket system with the loose fitting lid adequate?
I'd advise against using it. It is fine while the beer is producing CO2, but when fermentation stops you can then get air back into the fermenter. It is useful to leave the beer on the yeast for several days after fermentation is over.
Will transferring to secondary clarify the beer?
No. The beer will slowly clear over time as the yeast drop out. It can do this in primary or secondary just as well.
Others may disagree with me, but I would recommend you get the beer under an airlock. If you cany get an airlock on the bucket, transfer to the carboy (after it warms up and you have swirled the yeast into suspension).
For the ready-brew kits, how much time in primary? secondary?in bottles?
I'd recommend at least 3 weeks in the fermenter. You will get clearer beer, and it allows the yeast top clean up some undesirable by-products. 3 weeks at 70 F in the bottle is what is recommended (but you can always try one earlier).