[RPG] Does Synthesist eidolon need Weapon Training evolution to use weapons

As written, yes, you can combine that with a reach weapon.

3.X and PF rules are descriptive; that is, they generally tend to tell you what you can do. The RAW on the statement indicates that you can pick one attack - in the scenario you described, that'd be either 'manufactured weapons' or a specific weapon depending on the DM's interpretation of "one" - and the Eidolon gains another five feet of reach with it. Gaining reach in this fashion is fairly standardized, though this particular ability is not, and the flavor is admittedly a little odd.

"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.