Is there such a word as "foresitter" in English? Can I use it instead of "chairman"?

In the vast majority of cases chair(man/woman/person) is the best, even the only, word to use.

There are sometimes alternatives - you should consult a good dictionary (or two, the second being English-Russian) for their exact meanings and uses. These include president, presiding officer, secretary (though the other meanings of this are much more common without a qualifier, so it's best avoided unless the position has been given that title in English). You also have to consider the contrasting roles in the organisation: in some cases much of the chair's role is carried out by the speaker.

A generic leader could also work in some cases. In some cases English uses the closest word to the real title (we refer to the German Chancellor).

Foresitter is a direct translation that has no meaning in English. Other words starting fore- or front- aren't much help either.

It's rarely (possibly even never) wrong to use a long form such as chairman of the presiding council, which might guide you if you really need to avoid ambiguity in the title itself rather than just by context.

Foresitter would be almost incomprehensible to native speakers. There is a word foreman that could be used to translate a term for 'one who is in charge' if you do not want to use chairman or president. I do not, however, think it would be a good translation of political titles, because of the lowly connotations; a foreman usually receives instructions and translates them into work for others, sometimes (as in jury foreman representing his group to other authorities.

Contrary to the original poster's assumptions, "Chairman" and "Chairwoman" do not sound "weird" or "disrespectful" to my (American) ear. On the contrary, these words suggest "a person chosen (often by their fellow committee members) to control the flow of the committee's meetings". The chairman (or chairwoman) usually has a gavel, which they can bang to indicate that a meeting has started or ended, or that a decision has been made and it is time to move on. (Gavels are also used by judges and Speakers of houses of parliaments, which are similarly prestigious positions.)

"Foresitter" is not an English word. (My spell-checker does not recognize it, nor do any of the on-line dictionaries I checked.) "Foresitter" has some inappropriate connotations:

• "Fore" is the first syllable in "forelegs" (the front legs of four-legged animals)
• "Fore" is the first syllable in "foreplay" (things people do before having sex)
• "Sitter" is often short for "Babysitter" (someone who tends babies, or supervises children when their parents are not around).

To add to the other excellent answers, in some contexts 'head' may be appropriate in conjunction with what the person is head of, e.g 'Committee head', 'heads of government' (often used when describing a group including differing titles), and of course, as you yourself mention in the question, 'head of state'.