If I publish under a pseudonym, can I still take credit for my work?

Taking it that your asking about a nom de plume (aka "pen name") for reasons of conspicuousness, rather than for reasons of anonymity:

Yes, it is possible to publish under something other than your legal name. A nom de plume isn't that unusual in academia, particular for folk with common names who would otherwise be unfindable in literature databases.

You need to make sure that your legal identity is sufficiently tied to your nom de plume, so that there's not going to be any arguing about it.

Some people adopt double-barrelled surnames for their noms de plume: others add a distinctive first or second initial. As long as its sufficiently close to your real name, that should be sufficient.

Note that this will entail your nom de plume effectively becoming your name for academic purposes: it's what will be on your email correspondence, your web page, your conference name-badges, and so on. You'll just have to do a bit of tweaking with university administration so that payroll, legal, and travel arrangements are all in your real name, not your pen name.

I would like to use a nom de plume, but I would also like to use the publication to apply to graduate schools next time.

In principle, you could do this. You would need some way of demonstrating that you really were an author (nobody will believe you if you claim "Andrew Wiles" was just a pseudonym you used for your proof of Fermat's Last Theorem), but a letter of recommendation from your supervisor would suffice.

In practice, why would you want to? You would need to explain in your application why you had used a pseudonym, and I can't think of any explanation that would sound compelling. Even in the best-case scenario, this issue would be a distraction from the actual substance of your grad school applications, and it would probably hurt your chances of admission. Whatever you say, people are going to suspect it's because you are embarrassed by the paper or by your contribution to it, and that won't look good on an application.

I'm assuming here that your nom de plume is intended to hide or disguise your identity. If not, then it should be fine. Some people use different names professionally and socially, and this is OK as long as you are clear about it. (It may cause some confusion in your career, but it isn't considered an ethical problem.) If this is the case, then you should include a brief note of explanation, for example "I legally changed my name to Smith-Jones upon getting married in 2012, but I have decided to continue publishing under the name Smith."

No. You cannot simultaneously be anonymous and still receive credit for your work.

Theoretically, your pseudonym could get the credit, but you can't invite a pseudonym to be a collaborator, or to give a lecture, or to apply for a grant, so I'm not sure how useful that would end up being.