A modified version of two way DFA
To answer your first question, see: Burkhard Monien,
Two-way multihead automata over a one-letter alphabet.
RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications (1980) https://eudml.org/doc/92118

".. we show that for languages over a one-letter alphabet two-way automata with $k+1$ heads are more powerful than two way automata with $k$ heads."

This improves over a similar result of the same author and others for two-letter alphabets, one-way automata, or stepping from $k$ to $k+4$-heads, see the paper.

Also an interesting quote "It is wellknown that SPACE(log n), the class of languages acceptable within space bound log n, is identical with the class of  languages acceptable by two-way multihead automata".