1.) Yes
2.) You only get one from the quest. To get more you need to connect your game to one of the Let's Go titles for the Nintendo Switch and you'll receive an item called the Wonder Box. This is basically a Meltan only incense and lets you catch 9 or 10 per use. There's a 7 day cooldown on the box where you'll need to resend something to Let's Go to use it again after said cooldown. One Switch game can connect to multiple Go accounts, so if you know someone with it you can borrow theirs to get the box.
Meltan
(1) Is Meltan genderless?
(2) Can you only get 1? In other words, does this request or end like Mew, etc.
Answers
you can get an additional 10 per week (so 84 candy (91 if you meltan swap and release the ones you get in trades) if you do special trades and pinab all of them, then release the additional 3 you get from the box as you can only trade one meltan daily)
It's possible to get those additional 10 weekly if you have lets GO (and a compatable phone, as I learnt the heard way wit my phonn not being able to connect to my switch)
Small correction - the item is called a Mystery Box.
From what I understand, it's easy to get. One of my student collaborators walked me through the process with his switch; it was trivial from my end, and I got the box in trade for an Alolan Eggy. After opening it, it closes in a half hour (time enough to get 10 encounters), and then you have to wait a week before transferring another Pokemon to the switch (which i'll do next Monday). After doing so, it's enabled once again, and you can catch another 9-10 Meltan.
Why 9-10? It ought to be 10, but there's a quirky bug. As far as we know (we being a few people asking questions about this and posting here), if you have the box active during the change of hour, sometimes it messes up and gives you a different encounter once. I got 9 Meltan and one Gulpin, caught at 9:59. The next time I open it I'll do so sometime between 5 and 25 minutes after the hour.
Final hint - if you have regular access to a switch and anticipate being able to transfer regularly, it makes sense to look at the weather reports. Since Meltan is Steel, it's buffed by snow. Once mine is activated, I'll look at the seven day forecast. Any snow in there, and that's the day I open it, to get buffed Meltan (higher levels, higher IV floor). The few days lost are not a big deal - eventually we'll all have hundreds of them if we want it. But powering up a stronger one, at higher level, is a big deal.
It doesn't even need to be at the change of the hour, people have reported only getting 9 Meltan when they started at like 12 after. Theory is it's an incense bug that causes the local spawns to override the "incensed" Pokemon (in this case Meltan) that's never been pointed out before since regular incense spawns are randomized. With this working as a Pokemon-specific incense it's easier to notice.
It's also different per account. Two people can open a box standing side by side and one person can get shortchanged while the other can encounter all 10.