I think someone cheated in my previous game

You were playing CrazyHouse; that's why there are pieces on the left side of the board. You can drop pieces of your color on the board instead of making a move.

https://www.chess.com/article/view/how-to-play-variants-on-chess-com

You were playing the crazyhouse variant of chess.

There was no cheating involved at all.

This said, please do not accuse other people of cheating in public.

Standard advice below:

To discuss the matter of cheating further, or to learn more about cheat detection, please join the Cheating Forum: https://www.chess.com/club/cheating-forum

Chess.com takes cheating very seriously: see https://www.chess.com/article/view/chess-com-fair-play-and-cheat-detection,https://support.chess.com/customer/en/portal/articles/1444879-fair-play-on-chess-com-what-you-need-to-know and https://www.chess.com/blog/DanielRensch/cheating-on-chesscom.

Cheaters are banned when they are detected.  There are various tools that are used to keep them from returning without permission; these tools are familiar to anyone who has run an online forum.

However, discussions of cheating in the main forum is not permitted, as it tends to become very heated and result in lots of public accusations, many of which are completely unfounded; you might be surprised how many cheating reports we get about low-rated players after a game that was a complete blunderfest. If all these reports were aired in public, it would just cause an ugly flame war. Therefore, I am locking this post.

To report a suspected cheater to the Support team, please select Report Abuse under the Help menu, or use this link: https://support.chess.com/customer/portal/emails/new. All reports are investigated, even if there is no immediate visible action.

He summoned a rook from nowhere?  If I could do that just once a (regular) game, I'd be a GM.

"Someone" ... so was it you or your opponent? Because both are valid ways of interpreting what you said.

this forum should be in chess 960 or other variants.