Hey, I'm a current student (just finished GCSEs, starting sixth form in September); hopefully I can be of some help to you
So from your results you meet the minimum entry requirements which is great along with the requirements for the individual subjects
It's a great school (slightly biased I know) but both me and my brother have went to Latymer and we can both agree the teaching is top notch, and the people are amazingly diverse, intelligent and kind. Sometimes we get a reputation as a hothouse which forces you into incredible work loads and this isn't true from what I've seen over the past 5 years. From the results you've given you should excel.
If you know Russian proficiently it will be a good asset, especially since you're doing the subject for A-levels, however I don't do Russian so I can't advise you greatly on that.
The main problem I see is that you present 1 situation in which you're only taking 3 A-levels at school (Russian, History, Maths and studying Classical Civilisation at home) and from experience the school doesn't like this that much. Self studying seems to be one of those things reserved for people who have shown exceptional talent during previous years in this subject and can justify it (eg mathematicians taking A-level Maths in year 9). The school will make you take 4 AS-levels at least, for example Russian, History, Maths and FM. Whilst taking 5 AS-levels at Latymer is not by any means impossible, it is quite rare: I would estimate about 3-5 people take 5 AS levels per year and, from personal experience, they do a fairly thorough review as to whether you'll be able to handle the workload etc etc.
Again, though, if you can prove that you can handle it the school are more than happy to help in any way, our Latin department is top notch and one of the teachers is a personal favourite of mine and teaches Ancient Greek to enthusiastic students and if you were to ask he would be able to give you help on anything Greco-Roman.
I haven't gone through the external applicants process (obviously :P ) but from what I know there are no formal tests. You need an A/A* in the subjects you choose to study (usually an A* for external applicants from what I've heard) and you have interviews with subject teachers for the A-levels you want to take as well as the deputy head teacher. I do not think there are any internal exams which you'd have to take
Latymer is a great school; both the teachers and pupils are lovely and there's a huge amount going on at any point so you can take your picking. Sorry for the digression, I hope this has been of help and if you have questions shoot away XD