Welcome new pilot and thanks for volunteering! I am sure pilots will chime in and share their experiences with you. A few things to consider:
1. Health condition, age, and breed of an animal with regards to altitude.
a. Puppies really should not be flown if they are under 3 weeks of age but that being said, this has been done but with very slow ascent, descent and lower altitudes 5k and under. Their lungs are not developed well enough yet to handle extreme pressure. Many times rescues will risk sending younger, if the pilot is willing to transport, because if not they will be euthanized by the end of the day.
b. Pregnant dogs close to term may go into labor from extreme pressure as well, just like people.
c. Snub nosed dogs and cats already have some difficulty breathing. Add to that pressure, heat or cold, and it is more difficult. You can help with the temps by using covered ice packs or warm covered bread warmers, but only the pilot can control the altitude.
d. High heartworm positive dogs have difficulty breathing , minimum heartworm positive shouldn't be an issue. It is not recommended to fly dogs who are being treated for HW's unless the vet gives the ok. All animals should be flying with current health certificates so veterinarians should be the ones making the final decisions. Ask the sending rescue questions about health conditions as well for any animal you consider helping. Of course ultimately it is always the pilots call!
A good rule of thumb for general purposes is if you need oxygen, the animal will too!
Something else to point out is that though PNP does not have it's own call sign, you are permitted to use the Compassion call sign because we are a member of Air Care Alliance, should you so desire. Always good to put rescue animals in your flight plan if you are making one.
Here is a link to a previous discussion on altitudes:viewtopic.php?f=4&t=1674&p=17148&hilit=altitude#p17148
Pilots, can you please chime in?