You can say it's neutural. But it is NEGATIVE at some degree, showing your impatient, anger, and disappointment at something/somebody. To let you understand this I'll show you an example. 老板：（对职员）我让你干的活儿你完成了没有？ Boss: (To staff) Have you finished the task I told you? 职员：啊……那个……快了…… Staff: Ah...it's...about to finish... 老板：什么？还没做完？你怎么搞的！ Boss: What? Not yet? What's wrong with you? 职员：这个……我是想完成的……但是…… Staff: No...I don't mean to...but... 老板：算了！我找别的人来做！你走吧。 Boss: Well...I'll find someone instead. You don't have to come tomorrow. In my experience, Chinese use "算了" to express their disappointment at something/somebody out of their expectation, and have to give up the original plans. So next time, if something is out of control, or somebody let you down and you don't expect him/her can make it back. You can use "算了！" to express your thoughts. Hope it helps.

it's neutral. when somebody insults you and you say "forget it" means you don't want to fight back at him which can show your self-restraint sometimes. and if you insult others and then say "forget it" means you don't want to continue abusing others and let them go.

depends on the situation

negative or positive,it can depends ur tone sometimes it appears so arrogant e.g. A：我不明白你的意思。 B：算了。 it sounds like NVM here

It depends. negative: "算了，算了，我还是找别人来做这件事吧。"　it means forget it and expresses your impatient. neutral: "你想帮我？算了，我不想麻烦你。" it means that's ok