Sorry for the delay. RE the Kodokan, I wouldn't know (most national orgs are part of the IJF, which is technically seperate from the Kodokan). Deadlyknuckles seems to have some solid advice.
Re Beginners starting Randori, that happens in every Judo club I have been to and heard about. It is part of the training. Generally it may start from the ground (ne-waza) or may be "koka" randori (where even the most minor score results in the randori finishing), but it is a common feature. It is part of the appeal for some people (i.e. getting stuck in straight away). Some basic tips from Judo noobs would be to match up with the high ranks (they usually have no ego and are happy to let you try and throw them), stay with people the same weight at the start, and avoid people who have lack of control/esteem problems on the mat (the kind of person who tries hard to slam you to make themselves feel better). The more work they do on breakfalls the better.
It really depends on the group though.
Overall the early exposure to Randori can be a good thing. It lets would-be judokas feel how very hard it is to throw someone who doesn't want to thrown, and to move someone who doesn't want to be moved. Talk to the instructor about this to see how he approaches randori and beginners.
I would be slightly worried that a Judo coach has a lot of problems moving someone heavier than themselves, especially when doing waza or kata. During Randori is a different story, but if they are experienced then there shouldn't be THAT much trouble.
Re grading/promotion... most Judo schools do it this way. Theory is tested by asking some technical questions, and the student is usually asked to display some movement (throw, hold, pin, armlock etc...). The randori session usually determines how slow/fast a student progresses though. In modern Judo, I have heard of people getting a 1st Dan in 2 years, while I have met a guy who after 8 years of grading randori still hasn't got his Shodan. It can go both ways.
In fact, this is the way the Kodokan do it. They hold monthly promotional tourneys:http://judoforum.com/index.php?showtopic=30169&hl=black
This method (grade determined by victory in shiai) is known as Batsugan.
For most mortals, grading this way it usually takes 3-6 years. There are exceptional circumstances though. In Japan, a student can to A LOT of Judo in week (due to quality dojo availability and judo being a big part of PE in the school/uni system).
Personally, I don't see anything wrong the this method. Some people can bluff through grades for a while, but the wheat gets seperated from the chaff in the end. If you know your Judo, are committed to training, have passion for it and can perform Judo well, you shouldn't have to wait a certain amout of time just to get a belt. If you know Judo in your mind, and you know it in your body and on the mat there and then, you deserve your rank. Alternatively, if you are going through the motions or are a hobbyist, it may take a while longer.
To paraphrase Jirgo Kano "Where there is effort, there is achievement"