Yes a Pure Evil can be suicidal. Examples include Joker who's shown to at times have suicidal tendencies, Owlman from the animated movie who planned to destroy everything who even when he failed let himself die, and Moriarty from Sherlock who ends up committing suicide due to believing it's a way to ensure Sherlock Holmes can't win. There are others but these three are the first to come to mind, some might commit suicide if they feel the alternate is worse be it prison or a fate worse than death. Provided they aren't doing it out of remorse or to protect others it's not automatically a disqualifying feature.

Most definitely.

depends on the reasoning for suicide - if it is due to tragic mental illness or being pushed too far, no.. however if it is for reasons that simply amplify their evil, yes..

a good example are suicide-bombers / kamikazees and mass-murderers.. they will keep killing and want to die but their evil is amplified because not only do they want their own deaths they want as many innocent people to perish with them as possible.

also evil by its nature is a morality very prone to the "if I can't have something no one else can" - thus a truly evil entity is equally prone to fits of rage and suicide if things don't go the way they wanted.. many evil entities would see themselves and the entire world / universe perish rather than allow heroes to be victorious.. evil is petty.

suicide is only sympathetic if it is committed out of sorrow and with the intention of solely killing oneself - when you try and kill off everyone else around you it is no longer sympathetic in the slightest.. much is if you have DIssociative Disorder, it is sympathetic if the disorder harms mainly yourself but if you end up forming personas that kill, torture or generally ruin everyone around them it is no longer sympathetic.. (that's why some mentally ill qualify as villains, despite being ill they lose sympathy if their actions become heinous enough).