"Contract with the Abyss" is used as a generic substitute for Ritual Spell Cards like "Black Illusion Ritual" and "Contract with the Dark Master". You cannot use "Contract with the Abyss" to Ritual Summon a non-Ritual Monster like "Dark Magician".
"Cost Down" will not reduce the number of Level Stars needed to perform a Ritual Summon, even if used with "Contract with the Abyss".
If you Ritual Summon a Ritual Monster using "Contract with the Abyss", it is considered properly Summoned, so you can use "Monster Reborn" on it afterwards.