If, say, during the process a client, who is accused of committing a murder, admits that he did it to the lawyer and demonstrates some undeniable proofs, like video of him committing a murder in a private conversation, but asks the lawyer to prove that he didn't commit it and the lawyer follows the demand, withholds the evidence and lies and makes his/her arguments to the court that his/her client is innocent. If all the aforementioned actions are disclosed (especially the part about the evidence), are there any problems that may arise for the lawyer?
If an attorney knows that his client intends to or does engage in unlawful conduct during his representation, things get complicated for the lawyer who will presumably know this stuff. It would be illegal for a client to testify falsely, so if the client intends to perjure himself, the lawyer should diligently act to counsel to not lie under oath. There is no law requiring a client to confess or provide incriminating evidence (in lieu of a subpoena), so if the client plans to just sit quietly and let the attorney do the talking, then the lawyer is doing his job and he will not be punished, even if it is somehow discovered that the client is actually guilty.
The lawyer might have advance reason to believe that the client intends to commit perjury, so might not allow the client to testify. That happened in the case People v. Johnson, 62 Cal. App. 4th 608. Johnson's conviction was overturned because he was denied the right to testify in his own defense (Rock v. Arkansas, 483 U.S. 44). The Johnson court discusses various options for resolving the client-rights vs. attorney-obligation conflict, including the "narrative option", where the attorney does not ask questions, but rather allows the client to freely narrate events, and then the attorney does not rely on any false testimony. The court concluded that the narrative option is the best resolution of these conflicts. Disallowing client testimony is a denial of the right of the defendant to testify.
Another possibility is that, like Lowery v. Cardwell, 575 F.2d 727, a client might surprise the attorney with false testimony. In this case, the attorney sought (in chambers) to withdraw from the case (denied), and then did not further question the defendant. While the attorney's motivation was ethically admirable, the court found that it was prejudicial against the client, concluding that the attorney's action "amounted to such an unequivocal announcement to the fact finder as to deprive appellant of due process".
If an attorney were to encourage a client to commit perjury, they could be disbarred. But generally, the rule is that it is the prosecution's duty to prove that the accused is guilty, and not the defendant's duty (or the defendant's attorney).
"Withholding the evidence" in this situation is the lawyer's clear duty; any conversation covered by attorney-client privilege cannot legally be repeated, much less provided to the prosecution. Arguing for an acquittal is also a clear duty, and it can be done whatever your personal opinions are; for example, you can point out that the prosecution have not proved their case beyond reasonable doubt, and your client is therefore entitled to be acquitted, whatever the jury (and you) may think of his likely guilt.
Lying should never be necessary; since saying "I believe my client is innocent" is not a legal argument, many courts do not allow the practice. A lawyer's opinion of facts is no more reliable than anyone else's, and much less so than the jury's.
'Proving the client is innocent' is of course impossible if he is guilty, but bringing forward misleading evidence is a very difficult area of legal ethics; it may be a duty to put forward the evidence of a witness who honestly believes he can provide an alibi (even though you know he must be mistaken) but suborning perjury to put forward a witness (including the defendant) who you believe will lie.
One of the advantages of the barrister/solicitor divide in England and Wales is that a solicitor can continue to do his job even if he knows you are guilty; a barrister would almost certainly advise you to find some other advocate (for the reasons user5726 mentions among others), and if you refuse or do not have the time or resources, can continue to represent you under specific limitations laid down by the Bar Council (though he cannot, of course, disclose the limitations or the reasons for them till after the case).
No. A lawyer would be disbarred if he failed to defense a client even if he knew that client was guilty. A criminal defense lawyer is require to make "arguments to the court that his/her client is innocent."
In a criminal action, there is no affirmative duty to deliver or produce evidence, although the steps taken to withhold evidence could conceivable violate the law (e.g. it would clearly violate the law to kill someone who is a witness who could testify against a defendant and a lawyer who ordered that would be criminally and civilly liable).