How is Gendry supposed to fulfill Arya's request?

What is it: We'll find out soon enough, but it looks like it might be a dart similar to those used by Byzantine infantry, with some kind of thrower similar to an atlatl, which enables them to be thrown further.

It's different to the classic atlatl design, which is used with full-length spears, whereas this looks more suited to throwing smaller projectiles, which would allow someone like Arya to carry more darts and take out more undead. The dart or short spear would sit in the hollow part and the user could use an overarm cricket-bowling motion, flinging the dart forward with more force than would be possible using the arm alone. The small size doesn't sacrifice effectiveness given the effectiveness of even small wounds from dragonglass.

But we're speculating and that's just one possibility. There is some more discussion and other possibilities (such as a spear that splits into two one-handed weapons) under this Movies & TV question, "What is Arya's weapon design?", including some teasing quotes from the show's weapons designer. We'll find out soon.

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Why doesn't she provide a complete schematic: Smiths are skilled tradespeople, not hammer-monkeys who need precise instructions. They're the ones best placed to choose the type of wood, get the weighting and length right, etc etc.

There are few things more annoying for a master craftsman than a client who thinks they know your job better than you do.

Also, keep in mind that most smiths, and most of their clients, are illiterate - it's absolutely normal to ask for something to be made based on an approximate idea and size, where the skilled smith then uses their judgement and experience to work out the details. Sometimes the client may have someone like an architect who is giving precise direction, but for most commissions this won't be the case.