Looking at cantrips, a forest gnome gets two right out of the gate; Minor Illusion and Speak with Small Beasts (which refers to Beast Speech, which refers to Speak with Animals -- if accurate, I'm still going to limit it to tiny creatures).
Speak with Small Beasts isn't a cantrip, or even a spell. It's just an ability that forest gnomes have. It's also technically unrelated to the speak with animals spell, though a DM would be well within their rights to say that the interactions you can have with an animal using Speak with Small Beasts are limited in the same way as described for speak with animals ("The knowledge and awareness of many beasts is limited by their intelligence, but at minimum, beasts can give you information about nearby locations and monsters," etc.).
Looking at spells, at 4th level a warlock would have two 2nd-level spell slots and five slots for "known spells" (which, I gather, can only be swapped out when you gain a level).
Judging by your list of "Spells" and "Spells known" at the end of your post, you seem to have misunderstood the meaning of "spell slots." Spell slots represent how many non-cantrip spells you can cast before you need to recover your slots via resting. At fourth level, you would know five spells of first and/or second level, but you would only be able to cast one of those spells and then cast either that spell again or cast another spell you know, after which you would only be able to cast cantrips until you took a short rest. Also, eldritch blast, being a cantrip, is counted under your "Cantrips Known" rather than "Spells Known", so you need to fix your Spells Known by adding in another first- or second-level spell.
I'm not quite sure about the difference between invocations and cantrips ... Mask of Many Faces (which, it seems, might as well be a cantrip?).
Some invocations are like cantrips in that they let you cast a spell without limit, but the spells you cast via them are not actually cantrips. For example, Mask of Many Faces lets you cast disguise self, which is a 1st-level spell, and it will be treated as a 1st-level spell by any effects that care about spell level.
Most invocations are just constant static effects or give you non-spell options; those are definitely not like cantrips.
for Hit Points that should give me 8 (base) +2 (CON modifier) + 3x5 (fixed increase from each level above 1st) = 25 HP.
Not quite. With a Con score of 12 or 13, your Con modifier is +1, and you add it to your HP on every level, so your HP at fourth level would be 8 + 1 + 3×(5+1) = 27 HP.
Also, you can't get cure wounds through Pact of the Tome, as that spell is 1st-level rather than a cantrip (Cantrips are effectively "0th level").
Everything else looks accurate, though I do note you failed to specify a Warlock Patron (Archfey, Fiend, or Great Old One). You have to pick one when you create a warlock.