It really depends on the circumstances, the manner of death, the agency handling it, as well as their training, policies and procedures.
Generally speaking, when a suicide call goes out involving an obvious death, you'll have medical, fire and police respond. If you're freshly hanged for example while still warm without discoloration and death is less obvious, they're more likely to cut you down and try life saving measures, however if you are there with a big hole in your head and its obvious that you're gone well... some might feel for a pulse, most won't.
If its an obvious death involving suicide, they'll document the scene, question you and remove the body. If its a questionable death involving suicide, they'll still do the same thing essentially. Its the job of police to investigate and collect intelligence, however they cannot compel or force you to answer questions, and honestly you never should answer questions either without an attorney present, even if you've done no wrong.
As a spouse, if anything seems possibly amiss, you're automatically the prime suspect. They'll want your DNA sometimes, which you should decline without a warrant. They'll likely want to photograph your body for any signs of injury, which you should decline without arrest or warrant. If it involves a gun, they'll likely want to swab your hands for residue, which you should decline if there is no warrant. They might want your clothing which you should decline to provide, without a warrant.
We had a situation out here where a guy was out of town, hadn't heard from his girlfriend while he was away. He came home, she was dead and had obviously been dead. Police come along, he had a solid alibi and the death was very in line with a suicide. Well... police still tried to press him anyways.
They ridiculed him when he wouldn't answer their questions, said that exercising his rights was suspicious. He then got an attorney to advocate for him so he wouldn't fall prey to the verbal judo of police that can at times find an innocent person in cuffs. In the end he was vindicated, but he was wise to not only know his rights but flex them as well.
Just remember when it comes to police, even if they try to infer or accuse you of wrongdoing, its not your job or obligation to aid them in their investigation nor prove your innocence to them, its their job to investigate and their burden to prove you've done something wrong. Its so easy to talk yourself into cuffs these days.