Can someone who lives in "Halachic" Eretz Israel visit a part of Israel that is not part of its Halachic borders? For example, IIRC, people in Eilat keep two days of Yom Tov. Does that mean that a person from Tel Aviv would be forbidden from visiting Eilat (unless it is for business, learning Torah :) etc.) ?
Rabbi Aviner has stated clearly that Eilat is part of Israel, and living there fulfills the mitzvah of Yeshuv Ha'aretz. (Question 2 in the link)
Rav Wosner(zatzal) and Rav Yitzchak Yosef (shlita) paskened it is not part of Israel according to halacha, and one may not go there strictly for a pleasure trip.
Rav Elyashiv(zatzal) paskened it was questionable and therefore one should avoid: https://news.walla.co.il/item/884101
In order for any piece of land to have the status of "Eretz Yisrael" for the purposes of the doing the mitzvah of living in and conquering the land, two conditions must be fulfilled:
The land must be within what G-d promised to Abraham
The land must be under the sovereignty of Bnei Yisrael
Eilat fulfills both criteria. Thus, a Jew who lives in Eilat fulfills the mitzvah of living in Eretz Yisrael; also, he/she keeps one day of yom tov, and follows all other Eretz Yisrael customs (bircas kohanim daily, no tefillin on chol ha moed, etc.)
When it comes to the agricultural mitzvos, it could be that we need a Sanhedrin to sanctify newly acquired land in order to be obligated in trumot, maasrot, shmitta, etc.