Yes, it should work.Check out the datasheet for the IC on your board -> https://dlnmh9ip6v2uc.cloudfront.net/datasheets/Prototyping/TP4056.pdfIt takes 5V input to charge a LiPo up to 4.2V.
Yes, the first page of the datasheet in the Features section: ·Automatic Recharge ·C/10 Charge Termination Your LiPo probably is ok with 1A charge current, but it'd be best to check as LiPos do exist that require lower charge currents.