Is there a way to copy text style(font, weight, etc) in Photoshop?
Yes, there is. Ctrl + click (with + shift or shift alone, if there are many) all text layers in layers dialog to select them, then in character settings window, just set italic, font-family, weight, etc. For layer style, has been explained above.
Character Style and Paragraph Styles are designed for just this.
Much like layer styles, you simply format a text layer how you'd like it then click the little "new" icon at the bottom of one of the style panels. This will create a new style. When you then wish to format text to look like that style, simply highlight the text layer and click the style in the Character Style or Paragraph Style Panel.
Character styles will alter just the text - font, face, size, etc. Paragraph Styles will alter paragraph options - alignment, spacing, etc.