SCENARIO (For your convenience the scenario has been replicated for each applicable question. No details in the scenario have been changed, only the questions themselves.)

On March 15, 2020, a woman (Victim)is shot and killed in the elevator of a posh high-rise apartment in Dallas, TX at 8:15am. Police, through the course of their investigation, discover that Victim was in fact leaving the apartment of her married lover at the time she was killed. Police learn that Suspect was often violent towards victim and was frequently high from using and distributing illegal drugs. Suspect's wife was away on business for one week before and after the killing and Suspect had spent the evening before the killing in his apartment. Based on a valid arrest warrant issued on probable cause, police go to Suspect's 12th floor apartment to arrest him. Police do not knock but break open the door, only to discover Suspect is not home. Police search the home looking for Suspect, including inside a bedroom closet, where one of the detectives reaches inside and finds a gun tucked between two large blankets on top of a 6 foot high shelf. While leaving the building, Police spot Suspect entering the building and subsequently arrest him in the lobby for the Victim's murder.

One of the officers begins handcuffing Suspect while the other goes back up to the apartment to conduct a search where he finds an antique apothecary jar full of chloroform under the bathroom sink, near a towel with smudged lipstick markings matching Victim's shade.

At the police station, Police take a swab of Suspect's cheek and a hair sample without the consent of Suspect to match it with skin cells found under Victim's fingernails. Suspect's DNA matched the recovered skin cells.

Based on the facts above, can Suspect get a suppression hearing? Describe in detail why or why not.