Magic weapons and armour are not specified in the PHB. You can find them in the DMG, pages 213 and 152 respectively. Or in the Hoard of the Dragon Queen supplement, page 2.
With that said, yes, a +X weapon gives +X to both attack and damage. +X armour gives +X to AC (on top of the AC that armour of its type provides).
My tables: the characters know the underwater combat rules just as well as the players.
My interpretation--and this is just one man's thinking--is that anything in the PHB should be considered fair game for character knowledge. It's been my interpretation for decades, and has worked out well at plenty of tables.
The fundamental process is "GM narrates environment, player describes actions, GM narrates results." But we go deeper than that every time we play. We want to go deeper than that. We want there to be a world, not just the local environment. We want our characters to have histories, and backstories. This-all implies a lot of knowledge and experience on the character's part beyond just what's on the character sheet and what's been narrated at the table.
The PHB reflects all this assumed-common knowledge. At my tables the character knows her standing long jump is 6', even though she has no idea that she has a strength "score" of 12. My bard knows that he's only going to inspire someone X times per day, no matter how rousing the song. And all characters know that piercing works better than slashing or bludgeoning underwater, because they've spear-fished as kids, they've heard stories from their granddad, they've told and re-told in-universe myths and fairy tales that inculcate such knowledge. Jokes even. "Why'd the dwarf drown with his club in his hand? Because he couldn't beat his way out of a rain-barrel! With a club! Because it does bludgeoning damage!"*
In my experience, anything less than granting characters all the knowledge in the PHB quickly turns into a game of "mother, may I?" And that's not "the world's greatest roleplaying game."
* - I didn't say it was a good joke.
Best Answer
As a rule, yes. If it fits
The section you quote is specifically about selling equipment that you find (PHB, p. 144)
Rules for using equipment you find are found other places. Some significant rules are found in the following section on Equipment Sizes (PHB, p. 144, bold added).
So most of the time, if you find equipment and it is reasonable for you to use it, you can. A DM is the final judge of what would be "reasonable" to use. Note that this decision could be based on something other than a creature's size. For example, a bow made for an Illithid could have a grip too alien and strange to be comfortably used by a humanoid. Also, whether or not equipment has been damaged may be a factor. A DM might decide that armor worn by Orks is usable if they were disabled by a sleep spell, but not if they were killed by a mighty blow from a warhammer. Armor that usually would fit might become too dented to be worn after you kill its occupant (although a DM might then have to bend over backwards to explain why your own armor is wearable after you've succeeded on three death saving throws).
However, some DMs may ignore any of these concerns and declare that every piece of equipment you find can be used by you. It all comes down to what they consider "common sense."