Do I need Playstation Plus to play Star Trek Online on PS4

PS+ has limited support for + features for other accounts on the console.

PlayStation Plus benefits available to other non-subscribing users on your Primary PS4:
Online multiplayer (subject to any parental control settings on sub accounts)
Play downloaded games purchased with PlayStation Plus discount
Play downloaded IGC games

PlayStation Plus benefits not available to other users on your Primary PS4:
Purchasing PlayStation Plus discounted products from PlayStation Store
Online game save storage
Auto patch download
PlayStation Plus exclusive early access to game trial

So yes you can play PS+ games from the US account and play online, you'll miss out on several features like auto-cloud save backups, and you'll also have to log in with the US account with + to download free games (also EU/US PS+ games are occasionally different, especially on PS3). I'd strongly recommend getting PS+ on your main account rather than fussing with multiple accounts. The save data backup is nice, and the discounted games can make up for part (or all, if you buy a lot) of the subscription cost quite easily, as games are often an extra 25% or so off during sales with PS+.

Yes, local accounts are those that exist - are/have logged in - on the same console, whether created there or elsewhere.

The Playstation Plus subscription is tied to the SEN account, but that account can specify a single PS4 console as its primary PS4, which enables other (non-PS+) accounts to access everything on the console.

So any account using that console (the one specified as the primary by at least 1 PS+ subsriber) can enjoy the benefits of PS+, when using that console. That includes multiplayer.

I assume though that multiplayer is not available to local accounts if you are online and playing multiplayer on someone else's console. That part isn't exactly clear. But your (non-PS+)family members / housemates could definitely play multiplayer on your console with only your PS+ subscription when you're not using it.

This FAQ on playstation.com is fairly detailed:

PlayStation 4

You can activate one PS4™ on your Sony Entertainment Network account (this is known as your Primary PS4) and temporarily activate a second PS4 on your account.

PlayStation Plus benefits available to other non-subscribing users on your Primary PS4:

• Online multiplayer (subject to any parental control settings on sub-accounts)
• Play downloaded games purchased with PlayStation Plus discount
• Play downloaded IGC games

PlayStation Plus benefits not available to other users on your Primary PS4:

• Purchasing PlayStation Plus discounted products from PlayStation Store
• Online game save storage
• Auto patch download
• PlayStation Plus exclusive early access to game trials

PlayStation Plus benefits available to you on a non-subscribing friend’s PS4:

• All of your PlayStation Plus membership benefits are available to you when you use a friend’s PS4.

PlayStation Plus benefits available to your non-subscribing friend when you use their PS4:

• Play downloaded games purchased with PlayStation Plus discount
• Play downloaded IGC games

PlayStation Plus benefits not available to your non-subscribing friend when you use their PS4:

• Online multiplayer (subject to any parental control settings on sub accounts)
• Purchasing PlayStation Plus discounted products from PlayStation Store
• Online game save storage
• Auto patch download
• PlayStation Plus exclusive early access to game trials

When you sign into your Sony Entertainment Network on a friend’s PS4 the system is temporarily activated on your account. Please ensure that you correctly sign-out of “PSN” on your friend’s system to keep your account secure.

If your “PSN” account is inactive on your friend’s PS4, the system will be deactivated automatically from your account and your friend will lose access to any of your downloaded content. This content can be deleted from their system hard drive by going to Image (Settings) > System Storage Management > Applications. Next, highlight the game to delete and press Options > Image (X button).

Source: http://faq.en.playstation.com/app/answers/detail/a_id/5030