Last question first. You can only charge a second generation Pencil from a compatible iPad; there is no external charging solution.
It is recommended that you keep a second generation Apple Pencil docked with your iPad whenever it is not being actively used. Doing so will ensure that the Pencil battery is optimally charged - and, importantly, is protected from deep discharge.
An Pencil set-aside, unused and/or uncharged for an extended period, will deplete its battery - leading to irreversible battery damage. As Apple Pencil is sealed for life, if the battery is damaged is fails, the only remedy is replacement of the Pencil itself.
Your Apple Pencil is exhibiting symptoms of a dead battery - that will not accept or hold any charge.
The tiny internal Li-ion battery is susceptible to permanent/irreversible damage through being left discharged for long periods. Even some “new” pencils can exhibit signs of failure out-of-the-box if they are “old stock”. It is essential that if you have an Apple Pencil that you charge it regularly - whether used or not - so as to protect the battery from deep-discharge.
If the Pencil Battery has failed, the only remedy is to replace the Pencil. If the battery has failed and your pencil is within its one-year warranty, you should look to having it replaced by your retailer or at an Apple Store.
