Ok, so before I start I think I need to point out that you need to talk to your doctor about your medication and exams. I graduate from my medical degree in a couple of weeks and although I can offer information, I think you actually need to seek advice from your GP. I would be skeptical of following medication advice from people on an open forum (as hypocritical as that is considering what I am about to write).
Do you have a follow up appointment with orthopaedics? I would question them directly in regards to recovery time as they are the experts. Everyone differs, although as you are young (I assume, given you are doing A-levels?) and most likely fit, then recovery time is generally less than an older patient with hundreds of other medical conditions that impact healing time. So I can't help with your first question, having never had an arthroscopy, however, I can provide some information to the latter question. Whether it is useful is another matter. Along with your GP, who will be a useful source of advice about your medication and examinations, it might be worth talking to your school too to explain the situation. At least that way they are informed and you won't be at any disadvantage for doing so.
Firstly, are you taking regular paracetamol? Paracetamol has very few side effects (unless taken in overdose, so definitely don't take more than the dose recommended to you by your doctor). If taken regularly, paracetamol helps to provide basic analgesia (pain relief) that can help to reduce the need for other, stronger analgesics that potentially have more side effects, such as your codeine and tramadol. So if you aren't taking paracetamol, go to your GP and asked to be prescribed some (they can generally prescribe a stronger dose than that bought over the counter).
Both codeine and tramadol are opioid analgesics. They belong to the same class of drug as morphine, although both codeine and tramadol are weaker painkillers. The advantage is that they are more effective at reducing pain than paracetamol but unfortunately they also have more side effects. It is difficult to tell how an individual will react to opioids. Side effects that MAY be relevant to your exams include: either a slow or fast heart rate, palpitations, hallucinations, vertigo, euphoria, dysphoria (unease or low mood, basically the opposite of euphoria), mood changes, dizziness, confusion, drowsiness, sleep disturbances and headache. There are many other side effects too, but just because they exists certainly doesn't mean that everyone will suffer from them.
The one thing to note is that these are painkillers, and as such, it depends how much pain you are in as to what you should do. They don't directly affect healing time but obviously have a considerable effect on your own comfort. I can't imagine doing an exam in pain would be that good for concentration, but if you are feeling like you are having difficulty with attention, memory or concentration whilst taking the drugs, it would be best to talk to your GP about the dose you are on.
So basically - talk to your GP and explain the situation about your exams as they may reduce the dose or change your medication.