Hi Ivan
It is me again. :-) In the long run if you really want to do two jobs there is theoretically nothing stopping you and after a few years experience working part time as a PT is quite acceptable. Search and rescue is often voluntary and is based on a combination of skills such as mountaineering and first aid. Most PT courses require you to complete some sort of basic first aid standard so that is fine. In our country there is a programme called a sports medic which physiotherapy students can do while they are training - on their weekends they attend sports fields and provide first aid to those acutely injured. But if you are thinking of training as a paramedic that is another degree in itself.
My advice would be is to concentrate on becoming the very best you can be at your chosen profession and initially really focus on that goal. Other schemes like you are suggesting can often be done in the weekend but don't let it get in the way of your studying and progressing. You should be thinking of working full time as a physio when you graduate - at least for a number of years. It is only this way that you gain the experience. If you goal is to be a top sports PT it is quite competitive out there now and you will have to have a leading edge over other physios working in the area. However when you are out in the field after a few years experience then do whatever you want - part time work is fine.
BTW - physio education is hard and demanding - don't think it isn't. and if you intend to get into a good school you are going to have to prove your metal to get in and you will have to work very hard. there are many much easier ways of earning a living and being better paid. People do physio because they love it.