The argument that there was no probable cause is very unlikely to hold up in practice. The law is very lenient to the issuance and execution of warrants.

  Warrants are favored in the law and utilization of them will not 
  be thwarted by a hypertechnical reading of the supporting 
  affidavit and supporting testimony. For the same reason, 
  reviewing courts will accept evidence of a less "judicially 
  competent or persuasive character than would have justified an 
  officer in acting on his own without a warrant." Courts will 
  sustain the determination of probable cause so long as "there 
  was substantial basis for [the magistrate] to conclude that" 
  there was probable cause.

http://caselaw.lp.findlaw.com/data/constitution/amendment04/..., citing United States v. Ventresca, Jones v. United States, and Aguilar v. Texas.

If you're interested in some of the intricacies of 4th Ammendment jurisprudence, http://www.law.yale.edu/documents/pdf/1994Fourth.pdf is a phenomenal article.

Shouldn't the person the warrant applies to be served and/or notified of the warrant and given a copy? I was under the impression that a warrant was a way to give police rights that they do not typically have according to civil rights. It's a civil contract that they will not exceed the rights they are given, so I'd assume he should be given a copy with an explanation of why his rights are being lessened temporarily.

Is that really the case? If the police is executing an illegal warrant, but comes across someone, say, murdering someone else, is that not considered evidence?

(This might very well be the case, considering the "Fruit of the Poisonous tree" doctrine).

It is, but there's a loophole: the prosecution can argue that the police inevitably would have discovered the evidence eventually. It's fairly easy to claim for something like a murder because hiding a body permanently tends to be fairly difficult, and arguing that the murderer would have successfully hidden the body is very close to an admission of guilt.

Edit: if the police interrupt and prevent a murder, the testimony of the murder victim is also admissible. I don't think the argument "if the police hadn't illegally searched the area, the victim would be dead and unable to testify" works.