[RPG] How many hp does a starting eidolon have

"Multiattack" refers to two different rules items here. One is a class ability. The other is a feat.

As part of the Eidolon class feature, the Eidolon gains a special ability called "Multiattack" when the Summoner reaches 9th level. This is a special ability, not a feat.

The text of that special ability is:

An eidolon gains Multiattack as a bonus feat if it has 3 or more natural attacks and does not already have that feat. If it does not have the requisite 3 or more natural attacks (or it is reduced to less than 3 attacks), the eidolon instead gains a second attack with one of its natural weapons, albeit at a –5 penalty. If the eidolon later gains 3 or more natural attacks, it loses this additional attack and instead gains Multiattack.

The special ability grants a feat (also called "Multiattack") if the eidolon has 3 or more natural attacks. Otherwise, the special ability grants a second attack.

Can the Synthesist's eidolon gain that feat? No. The Synthesist's "Fused Eidolon" special ability says this:

The eidolon has no skills or feats of its own.

Therefore at 9th level, when a Synthesist's eidolon gains the Multiattack special ability, the effect depends on the number of natural attacks that the eidolon has.

• If the eidolon has 3 or more natural attacks, the Multiattack special ability has no effect for the Synthesist's eidolon, since the special ability would usually grant a feat, while the Synthesist's eidolon cannot have feats.
• If the eidolon has less than 3 natural attacks, the Multiattack special ability grants a second attack, as proscribed in the ability text.

Does the Synthesis gain the benefit of this ability? Yes. If the eidolon gains a natural attack via this special ability, then the Synthesist, when fused with the eidolon, gains that attack. From the Fused Eidolon class feature text:

The synthesist also gains access to the eidolon’s special abilities and the eidolon’s evolutions. The synthesist is still limited to the eidolon’s maximum number of natural attacks.

The feats that increase your effective level for class abilities, such as Boon Companion, all have a clause that says "...to a maximum effective XXX level equal to your character level." A single-class character can't use them to exceed their actual class level. These feats are solely meant to take a little of the sting out of multiclassing while still allowing you to Do Your Thing.

There is no such feat for summoners published in Paizo products, though there may be one in a third-party product. The summoner is already a very strong class, and while adding it wouldn't break the game, adding one is not necessarily a great idea. Since eidolons are the bulk of the power behind the class, at feat like that would allow a character to take 4 levels in any other class with very little downside.

If absolutely had your heart set on a feat like this (and you could talk your GM into it), you could do something like this:

Boosted Eidolon

Benefit: The evolutions you can select for your eidolon are calculated as though your class were 4 levels higher, to a maximum effective summoner level equal to your character level. If you have more than one eidolon, choose one to receive this benefit. If you replace the eidolon that has received this benefit, you may apply this feat to the replacement creature.