I think the best way to understand this is:
再 X usually followed by 也 / 还是
- no matter how X
- as much as ... is X
It is used in concessive clauses more or less equivalent in meaning to 不管 / 无论 ... 多少 / 多么
你再努力也没用 (similar to 不管你多么努力...)
No matter how hard you try, it's useless
As much as she dresses beautifully, Zhang Jie still likes another girl
No matter how expensive this TV is, I still intend to buy it.
About your questions:
Can I use 再 like this in replacement of 无论?
Yes, but you have to rework the word order a little:
无论他吃多少药，病都没有好 => 药他吃得再多，病也好不了
Actually in this specific example I think the second sentence 病都没有好 isn't a perfect fit anyway. Either 无论～多少 or 再 express what we could call a prediction. When I say "no matter how much drugs you take", I'm implying that I don't know how much you will actually take, but it simply is not relevant. After that sentence you use 病都没有好, which has a past connotation (没有好 = hasn't gotten any better). So using a prediction with a perfective action is weird.
So in my translation I changed the perfective 病都没有好 to something else 病也好不了 (or 病也不会好) that has a compatible aspect.
How does 再 (no matter how) work with 即使 (even if)?
即使 appears as emphasis; I think it's redundant. It makes sense in this sentence because the first clause is long, so it introduces the concessive clause without waiting for 再. You could write the same sentence without 即使。
Another way to understand it, in a more prosaic translation, could be:
即使 ... 再 X
Even if one has plenty of skill and experience, without sense of responsibility toward their own work, they'll hardly do a good job.