I would suggest thinking in terms of space / location / spacially / room, when trying to build sentences with "frei sein". Also do not translate it to "available" but to the simple "to be free". That fits the meaning far better I think, especially in everyday-speech.
frei sein
frei sein = to be free
- as in: available, not in prison/shackles/tied up/in a cell/etc., free to do, allowed to do, feeling unbounded/free, free to act/decide what to do with your time. - "Ich bin frei zu tun was ich will." ("I am free to do what I want")
- and also: unoccupied, clear, devoid of people or obstacles (as in: the road is free/the road is clear, the tube is free/clear/unclogged - "Die Straße/das Rohr ist frei.")
If someone/something "ist frei"/"is free" he/she/it has "the space" (the room for decisions or actions) in order to accomplish "something" (also in the sense of a chair providing space to sit).
Sometimes also tied to money/prices. More common would be "gratis" or "kostenlos" instead of "frei" though:
- "Does this cost something?" - "Kostet das was?" - "No, it's free." - "Nein, ist frei." (colloq.)
- "Is there an admission fee? Does it come with a cost/price?" - "Nein, es ist frei (für alle)." (access or entry is granted free of charge (for everyone))
- But: Asking "Ist es frei?" - "Is it free?" does not work in the context of pricing: (wrong!)"Ja, es ist frei/kostenlos.". The case is more like: "Is the chicken free?" - "Yes, the chicken is free." - "Ja, das Huhn ist frei!" (omg, the chicken runs around out of the cage, not: it's free to take one).
Asking for an appointment: Even if this is related to "time", the thought is more like asking if someone has a "slot"/a space free in the calendar. Therefore "ist frei":
Think like: "Is there a space free for me in the doctor's schedule?" - "Is there a time available for me?" - "Does the doctor have time for me?" - "Is the doctor free for me?"
And say something like:
- "Ist da noch (ein freier) Platz im Terminkalender?"
- "Ist noch ein Termin frei?"
- "Gibt es noch einen freien Termin?"
- "Gibt es noch einen Slot (für mich) im Kalender?" (business speech)
- "Ist der Doktor frei (am Montag/um 12)?" - is okay but kinda old-schooled. Ask for an appointment or slot, instead for the doctor himself, except you want to invite for dinner or something.
- "Ist der Doktor verfügbar (=available)?" - is odd. You might use that one on the phone if you have the clerk and you want to speak to the doctor. But more likely in this case would be "Ist der Doktor zu spechen?" instead of "verfügbar/available/frei".
- "Wissen Sie wann der Doktor verfügbar(=available) ist?" is odd. More likely: "Wissen Sie wann der Doktor da ist?" - "Do you know when the doctor is in?" (da/be there is asking for the days when he is in office)
- Other alternatives for asking for an appointment (everyday-speech): "Wann wäre der nächste Termin frei?" - "(Wann) Kann ich einen Termin bekommen?" - "Können Sie mir (bitte) einen Termin geben?" - "Ich brauche einen (neuen) Termin, bitte."
Asking about being occupied/taken:
- "Is there a space(seat) free at the table?" - "Is the (space at the)table occupied?" - "Is the seat(sitting space) available?" - "Is XY free?"
- --> "Ist dieser Tisch (noch) frei?" - "Ist XY (noch) frei?"
Concerning feelings or permissions, it gets a little fuzzy:
- "I feel (being) free." - "I fühle mich frei."
- "I am free (=out of prison)" - "Ich bin frei."
- "I feel free to taste one of those." - "Ich bin so frei und probiere eins davon." (polite) - "Ich bin so frei mir zu erlauben, eines zu probieren/zu nehmen" (is a little over-polite/old-schooled).
- "May I have one?" - "Sure, they are free." - "Klar, die sind für alle." ("sure, anybody can have one") (polite)
- "May I?" - "Darf ich?" (then doing something/a small gesture) (polite; except little kids repeating that multiple times: "Mom, may I"? - "Mama, darf ich, darf ich? Bitte, bitte, darf ich?")
- "Am I free to proceed?" - "Darf ich weitermachen?"
- "Am I free to chose one (freely)?" - "Darf ich eins (frei) aussuchen?"
In the last examples, note the switch to "dürfen"/"being allowed to do sth." instead of using "to be free to do sth.". Therefore: (wrong!)"Bin ich frei weiterzumachen?" or (wrong!)"Hab ich frei weiterzumachen?"
frei haben
frei haben = to have time / to have (free-)time
"frei haben" in german is more like a special case and is only tied to "time", not "space".
- "Hast Du frei?" - has a context like school or work: "Do you have school tomorrow? Or are you free tomorrow?"
frei haben = offtime / home time / work/school is over / I can go home / I am on holiday / I have free-time
- "Are you working tonight?" - "Nein, ich hab' frei." (colloq.) - "No, I'm free."
More examples with "frei haben" and more emphasis on "to have time":
- "Hast Du Zeit morgen?" - "Are you free tomorrow?" = "Do you have time tomorrow?" - "Yes, I am free" or "Yes, I have time (for you)." - "Ja, ich habe Zeit (für Dich)."
It can also be seen as:
- "to have free time" ("freie Zeit haben", time which is free, used rarely) and
- "to have free time" ("Freizeit haben", free-time, having time for oneself like in holidays, used often)
which might be the source of the confusion.
"Freizeit haben" ("to have free-time") and "freie Zeit haben" ("to have free time") is not ment as a time span with a definite start and ending, like in an appointment. It is more generally speaking "having time" of undefined length (the whole day). More like in: "are you willing to do sth. because you have this free time at hand?" ("Do you want it to spend with XY?"):
- "Do you want to come with us?" - "Yeah, I am free." - "Ja, ich habe Zeit." - "Ja, ich habe Lust mit zu kommen." - "Yes, I'd love to join in."
- "Willst Du Zeit mit uns verbringen?" - "Yes, I'd love to spend my (free) time with you." (both ways of meanings work: free time and free-time) - "Ja, ich möchte meine Zeit mit euch verbringen" (note that the free is omitted in everyday speech often und just time "Zeit" is used).
Note that "Zeit haben" in "Hast Du (mal) Zeit für mich?" - "Do you have time for me?" - "Do you have a minute?" implies seeking for comfort or someone who is "lending an ear". This is a very personal level. Therefore it is not used when asking for an appointment (doctor's or otherwise; even if you could argue that seeing a doctor is a very personal level).
Therefore:
- "Hat der Doktor Zeit für mich?" - "Does the doctor have time for me?" (you can use that, but it does not ask for an appointment)
- "Wann hat/hätte der Doktor Zeit (für mich)?" - "When has the doctor time for me?" (you could use this also to ask for an appointment (especially with "hätte") but it is very colloq. and a little sloppy, but far from being offending)
- "Wissen Sie, wann die Ärztin frei hat?" would ask for the time when the doctor is out of office, either this day ("Feierabend") or maybe also more in general having vacation time ("Urlaub") (but odd, because too personal). And (wrong!) for asking an official for an apoointment.