How to delete line(s) below current line in vim?

The delete ex command will work nicely.

:+,$d

This will delete all the lines from current +1 till the end ($)

To delete the next 2 lines the follow range would work, +1,+2 or shorthand +,+2

:+,+2d

As @ib mentioned the :delete or :d command will move the cursor to the start of the line next to the deleted text. (Even with nostartofline set). To overcome this we can issue the `` normal mode command. `` will jump back to the exact position before the last jump, in this case the :d command. Our command is now

:+,+2denter``

Or as one ex command

:+,+2d|norm! ``

To make this easier we wrap this all up in a command:

command! -count=1 -register D :+,+<count>d <reg><bar>norm! ``

Now to delete the next following 3 lines:

:3D

This command can also take a {reg} like :delete and :yank do. So deleting the next 4 lines into register a would be:

:4D a

For more information

:h :d
:h :command
:h :command-register
:h :command-count
:h ``

dG should work.
This means delete all rows until end of file from current cursor.

This will delete ALL lines below the current one:

jdG

Unfortunately that will move the cursor to the beginning of current line after the deletion is made.