There is no direct equivalent to "paste in front" or "paste in back" in Inkscape*.
Instead do Edit > Paste in Place Ctrl+Alt+V. This will paste the copied object at the top of the stack.
After pasting, change the order in the stack using the following shortcuts:
- Raise to top Home
- Raise PgUp
- Lower PgDn
- Lower to bottom End
If your document contains a lot of objects, this could quickly become confusing and tedious. So, a possible work-around is to Group an object Ctrl+G, then double click the object to enter its group, then Paste in Place Ctrl+Alt+V will take place within the group (on top of the object in that group), and from there End will move it to the bottom of stack within that group. Double clicking outside the group will exit the group. If you don't want the group, you could just ungroup it after you have done this using Ungroup Shift+Ctrl+G.
Alternatively, if you want to see the positions in the stack visually, open the Objects panel (Object > Objects).
Move a selected object up or down the stack, by clicking and dragging it, or use the buttons at the bottom of the panel, or use the shortcuts listed above.
*Note: If these functions are something you'd really like to see implemented in Inkscape, then perhaps add a feature request in the Inkscape forum. Hopefully if the Inkscape devs ever implement it, they might consider calling it "paste behind", rather than Adobe's attempt to butcher the English language by calling it "paste in back".