First of all, you shouldn't confuse actual gender with grammatic gender. Often they correspond, but not always. For example, a German for “girl” — Mädchen — has neuter grammatic gender (despite real-life girls are females).
The word стать is for actual gender. For grammatic gender we use the word рід.
As for nouns in Ukrainian:
Most nouns are bound to specific grammatic gender. For example хлопець “boy” is masculine, дівчина “girl” is feminine, дерево “tree” is neuter. (In these examples grammatical gender corresponds to actual gender, but that isn't always the case. For example: людина “human” grammatically is always feminine (irrespectively to actual gender of the human), стілець “chair” is grammatically masculine (which you can't say for sure about real-life chairs), одоробло “bulky thing, or awkward person” grammatically is always neuter (despite real-life awkward persons are males or females). Still, even if grammatic gender doesn't correspond to actual gender, such nouns still have immutable grammatic gender.)
Some nouns have no fixed grammatic gender: they can act both as grammatically-masculine nouns and grammatically-feminine nouns. For example, the word нероба “sluggard” can be used both as grammatically-masculine and grammatically-feminine:
Той лінивий нероба нічого не зробив.
Та лінива нероба нічого не зробила.
(Both sentences mean “that lazy sluggard have made nothing” — but note grammatically-masculine versions of той “that”, лінивий “lazy” and зробив “have made” in the first sentence and grammatically-feminine versions of та “that”, лінива “lazy” and зробила “have made” in the second sentence.)
Some linguists describe such nouns as having спільний рід “common/joint/mixed grammatic gender”, though other linguists describe them simply as allowing both masculine and feminine grammatic genders.
As I've said before, grammatic gender isn't obliged to correspond to actual gender, though for the nouns described in the current bulleted list-item it usually does (because if a noun allows usage in both grammatic genders then there's no reason to choose noncorresponding one).
Most profession nouns in Ukrainian are quite strange thing. In short, they are more “gender-plastic” than usual nouns (that're bound to specific grammatic gender), but less “gender-plastic” than typical grammatically-mixed-gender nouns (that allow to be used both as grammatically-masculine and grammatically-feminine). For example, the word директор “director”:
Мій новий директор наказав з'явитися на збори.
Мій новий директор наказала з'явитися на збори.
(Both sentences mean “my new director have ordered to appear at the meeting. Both sentences use мій “my” and новий “new” in grammatically-masculine form. But while the first sentence uses наказав “have ordered” in grammatically-masculine form, the second sentence uses наказала “have ordered” in grammatically-feminine form. The second sentence may look strange as some part of the stuff corresponding to the noun (мій, новий) is in grammatically-masculine form, while other part of the stuff corresponding to the noun (наказала) is in grammatically-feminine form — but that's a preferred way to use profession nouns like директор in formal speech.)
Most linguists also describe such nouns as grammatically-mixed-gender — but the fact is that they behave very differently from the nouns described in the previous bulleted list-item.
Most of the Ukrainian profession nouns (irrespectively to whether they have any feminine counterparts or no) belong to the 3rd category: when applied to a man — they're used in obvious way; but when applied to a woman in formal speech — the preferred way to use them is to masculine grammatic gender for preceding adjectives and and feminine grammatic gender for following verbs.
There is a growing tendency to use women-specific counterparts, especially in colloquial speech, e.g. not директор “director (either man or woman)”, but директорка “director (woman)”. Still, usage of many women-specific profession nouns in formal speech is controversial. Acceptance for women-specific profession nouns varies: some of them (e.g. вчителька “teacher (woman)”) are almost acceptable even in formal speech, some of them (e.g. стоматологиня “dentist (woman)”) are hardly acceptable even in colloquial speech.