For your specific character, you are correct.
However you left off the invocations (which can be quite invaluable and often act as a spell without needing a spell slot). I'll break this down specific to your character:
Cantrips: 2 base +3 from Pact of the Tome = 5 cantrips (+1 from Infernal Legacy covered below)
Spells: 4 known + Hellish Rebuke - these spells must be selected from the Warlock spell list. The Great Old one patron allows you to select from an expanded spell list which includes the spells listed under Great Old One on PHB pg. 110. This means you can select from these spells when you learn a new spell, not that you know them in addition to the ones you do select.
Invocations: 2 invocations, these do not count against spells known. There is no requirement to know the spell in advance, otherwise this would be addressed under prerequisites in the invocation. In fact, the Invocations have a different name, and you can infer that the name of the Invocation is the Warlock version of said spell, with the spell name under said Invocation being used as reference to it's capability. This is demonstrated through Invocations such as Armor of Shadows which is effectively Mage Armor without the spell slot or material component requirement. Another example is Book of Ancient Secrets, which besides letting you pick any 2 rituals from any class (wow!), it allows you to cast a lot of your Warlock spells as rituals so you don't have to expend a spell slot (which are invaluable as a Warlock).
Infernal Legacy: All spells known from this are separate and distinct from your spell slots and invocations. These do not use spell slots to cast but function on a specific recharge as indicated by the description under Infernal Legacy. This is separate and distinct because it is not a Warlock feature, but a Race feature and does not require spellcasting or pact magic in order to utilize (it's in effect the same thing as a Dragonborn breath weapon).
Technically when you give up an invocation in favor of another you give up all the abilities that go with it.
However, the book is silent on what happens with the ones you may have scribed in the book while you still had the invocation so that would fall under the decision of your DM. Keeping in mind that the invocation itself is what allows you to utilize the rituals in your Book of Shadows whether your DM says they are still there would really only possibly benefit a Wizard or someone with the Ritual Casting feat to copy them. However, you obviously can no longer scribe anymore nor cast the rituals in the book upon replacing the invocation.
Ritual Caster feat seems a bit iffy as well to allow access to those already scribed since it is the Book of Ancient Secrets that allows you to cast from your Book of Shadows as rituals and not from a standard ritual book which is procured with the feat.
Not official but lends itself to that conclusion as well.
Best Answer
If a feature lets you cast a spell, you use the casting time of that spell, unless stated otherwise. You just follow the general rules for the spell.
Note that mage armor has a duration of 8 hours and false life has a duration of 1 hour. Both can be cast well ahead of combat. Also beware that both invocations you list only allow the spell to be cast on yourself.