Kicks and body kicks would be OK, but I am not OK with taking punches to the head. Is this unavoidable in all arts?
Kyokushin Karate is also well known for forbidding head punches.
In most Kyokushin organizations, hand and elbow strikes to the head or neck are prohibited. However, kicks to the head, knee strikes, punches to the upper body, and kicks to the inner and outer leg are permitted.
There are striking arts with precisely the rule set you describe. World Taekwondo allows kicks to the head but not punches to the head. Punches to the body are legal. This is the competition rules used in the Olympics.
From the World Taekwondo rules document WT Competition Rules & Interpretation (Hammamet 04052018):
Article 14 Prohibited acts and Penalties
4 Prohibited acts
4.1.8 Hitting the opponent’s head with the hand
No, it's not.
You could pick up something that does not involve kicks or punches at all, like Judo or Sambo. Other alternatives are arts that typically do not involve actual hard sparring, like many styles of Kung Fu, Aikido or Karate. Jiu-Jitsu (not the Brazilian one) is usually pretty light on the sparring as well while still teaching you plenty of useful kicks and punches.
Apart from this: Sparring in MA is usually not all that painful and you will not suffer regular brain concussion or anything, unless you go to a crazy gym. Also you will wear protective gear at pretty much any point in sparring where head strikes happen and usually in competitions on amateur level as well.
Take a look at what happens at your local gyms and find something you are comfortable with. Personally I have gone through 20 years of various martial arts and suffered zero brain concussions and very little headaches so this stuff is definitely not mandatory.
With all things, it depends. Even in the arts above, and other arts that forbid punches to the head in a competitive aspect (ATA TKD also forbids hands to the head in competition, but allows kicks), many of them still teach hands to the head in self defense training.
There is a difference between competition rules, and self defense instruction. By definition, almost any modern competition will restrict moves to some degree (Such as groin kicks. That is a near universal no-no). Self defense rules should really have no restrictions, other than what is needed to practice them safely. And, there are many schools that teach competition style only, and do not teach beyond that.
So your best bet is to talk with your instructor about your concerns. However, even if it is allowed in training, it should be done in such a manner that it isn't a future health concern.