It varies so much from person to person and even animal to animal. I have 3 dogs and the two with shorter hair bother my allergies more than the longer haired one. I've never seen anything that says this is a medical fact but my sister's ex husband was the same way but a little more extreme. Living with them is fine overall but if I really scratch the shorter hair dogs and get a lot of hair and dander on my or if they scratch me I start getting hives in the area. Allergy medication helps and symptoms are usually gone pretty quickly.

With cats I react more like seasonal allergies itchy eyes, runny nose, etc after being in a house with them for more than an hour or two. I'd say symptoms can last close to 24 hours. 

he gets a contact rashand instant itchying minutes into a cat rubbing against him or sitting or carpets/sofas that cats have layed on..

there are times his eyes go puffy and red and runny nose... we recently re-homed our 3 kitties and that first week it seemed his allergies got worse. My sons contact allergies were getting more intense but once you are rid of the pets, there needs to be dialy vacuuming and mopping, washing rugs or sofas cats had slept on to properly clear the house 

one bedroom in our house remained locked and cat free fot during the day or my son would get so itchy and really bad red crusty rash from the cats. If inlaws came over i physically locked that room down because they would leaveit open and we’d have cat dander going in. I still vacuumed and mopped that cat free room daily

Small caution about pet allergies: you may not be allergic to your pet, but allergic to something in the pet's environment. I am not allergic to cats or dogs, but I am allergic to certain pollens. I end up with a red rash or welts on my skin from certain pollens.  If my cat or dog is outside and then carries this on their coats and it transfers to my skin: bingo, I have a rash and sometimes  itchy sinuses.
So for the cat I don't plant anything in  my own garden to cause a rash and I prefer to walk my dog along the beach areas. Spring time I am  particularly wary. Also hair brushing for both happens regularly.

Now to your questions. People react differently to the same allergens, but it is typically respiratory or skin breakouts with pet allergies. Not sure I came across people who are eating their pets (for symptoms like stomach pain or vomiting)... So its something you breathed in or settled on  your skin. Symptoms of allergies: sneezing, runny rose or eyes, post-nasal drip, itching or  swelling of the mouth / roof of the mouth, irritated sinuses, wheezing, coughing, shortness of breath, facial pain or congestion, rash, welts, itchy skin... Any breathing problems are serious and you may need to see a doctor.

Medication? Don't take anything without speaking to  your doctor - especially if you are on any medication already.  My own symptoms will be a skin rash and/or the feeling that I need to scratch  the inside of my mouth and nose. I normally self-medicate with Claratyn and the symptoms are gone within 3 hours. (I will wash off the area with the rash too because it is most likely their is some kind of pollen on my skin, even if I can not see it.)

If it is the pet, you can reduce the allergen (dander, hair, saliva) with regular vaccuuming, cleaning and possibly bathing your pet on a regular schedule okayed by your vet.  Hypo-allergenic pets? Not sure that this is anything but a pet shedding less. There is no "allergy proof" cat or dog.

One time I stayed at my sisters who has cats.  I drove 5 hours home, and had dinner with a friend who is a Dr.  Friend told me I was on verge of needing to go to ER.  Breathing heavily.  I use OTC, not perfect.  Not willing to get shocks. 

For me symptoms start with my chest closing down which can fairly quickly go to an asthma attack.  My nose starts bothering me, I have sneezing attacks and my sinuses close up.  And OMG, the itchy & burning eyes make we want to claw my eyes out. 

The onset can be as quickly as immediate to taking a period of time to come on.   If I am in a pet home and kids and pets are running around it hits me immediately.  All of the movement kicks up animal dander which is a killer for me.   If the home is kept clean, vacuumed and dusted regularly, and the pets and kids are kept under control I can tolerate being there for a longer period of time.  If I touch the pets (I love animals) I have to immediately wash my hands to avoid putting dander covered hands near my face.

My symptoms persist until I leave the pet environment.  If I do not leave that environment fairly quickly I risk progressing to full blown asthma and potentially bronchitis.

Just leaving is not usually enough though. I have to change clothes and shower to get the dander out of my own hair. If I don't, it drops out of my hair into my face and bronchitis is a strong possibility.

People who do not have allergies have no idea how miserable allergies can make a person.

Meds help but are by no means 100% effective. I take Zyrtec and Flonase daily and have for many, many years.  Though I feel allergies even while on daily meds, the meds help keep my nose open and reduce many of the allergy effects. I still feel like shit during my key allergy seasons but I can at least breath.

On a pet note.  Bichons don't bother me as much.  Bichons have hair Vs fur.  However, if a Bichon spends a lot of time outside and is not bathed daily they bother me too. I also have pollen, dust, and mold allergies.  Any pet that spends significant time outdoors is just a allergen mop and if they are not bathed obsessively they are worse than indoor pets as allergy triggers for me.

Allergies and asthma can be autoimmune conditions.  They are the first autoimmune condition that presented for me and it started for me when I was very young.  I am also a T-1 diabetic which is also an autoimmune disease.  While I worked and we lived in Qatar I worked with a geneticist that put me through extensive genetic testing and that I have an alleal strongly indicating autoimmune conditions.  Not surprising considering my allergies, asthma and T-1 diabetis.  I gave him a ration of crap about testing me to prove the obvious.  He and his wife and my bride and I became friends and would have dinner parties at our home or theirs fairly regularly. We were just starting deeper dive of genetic tests when we transferred to KSA.  I was his favorite testing lab rat.