Malleability: The Bitcoin wiki explains possible malleability, and we all know Mark Karpeles blaimed the malleability for his problems. As I understand, there was an update to Bitcoin Core to put a stop to this, but where are we at right now? How can malleability be exploited and in what way is it harder to exploit than before the recent update?
You cannot put a stop to this. Not without a hard fork. All they could do was to consider a certain transactions non-standard, but it does not prevent modified transactions from being mined.
Malleability is more of a term, a thing that developers should be aware of, rather than a problem.
If you know that it exists and what is concerns, it would not be much of a problem for you.
Unless you have a need to spend yet unconfirmed outputs - then the malleability can disturb you from doing it in a reliable way.
Double spending: How easy is double a successful double spend to create nowadays? Does the Bitcoin Core still broadcast double spends that come in later? Can it still be done easily by importing the key to two clients and then just spend the bitcoins a second later?
Bitcoin is a technology that was invented to prevent double spending, so I really don't know what you mean.
You just need to wait for enough number of confirmations to be proportionally ensured that a certain transaction was not a double spend.
In most cases 1 or 2 confirmations is enough, but the industrial standard ever since seems to have been 6 confirmations.
In both the cases remember that Bitcoin was never meant to be a fast payment processing network. The protocol has a potential for extinctions that would facilitate (e.g. off-chain txs), but these days it's more of a theory rather than a reality.
Anyway, 99% of bitcoin users can easily live without spending unconfirmed outputs and each of us is totally aware of the fact that if we deposit coins somewhere they usually get credited after 6 confirmations. The reaming 1% (of mostly newbies) will sooner or later adjust to these rules, having not much choice anyway.