I don't know the fine details of karmagen, so I don't know if you can afford everything.
1.) In combat healing isn't really possible in SR. Out of combat healing is quite viable and reasonably quick (on the order of < 1 minute). The two major ways to heal is First Aid and the Heal spell.
First Aid needs lots of dice. Since it's a threshold 2 test, you need 3 hits to heal that first box of damage. Every additional hit heals another box. Thankfully rating 6 first aid kits are cheap so you'll want to grab one for +6 dice. The first aid kit also can replace your skill (or provide it). I suggest using the Augmentation rule that your max boxes healed is equal to your skill or the rating of your medkit whatever is higher. Otherwise, having only a few ranks of first aid is either useless or worse than useless. You'll want a good logic score as well.
Heal uses your spellcasting roll so all you need is a good spellcasting roll. If you're a mage / mystic adept, you'll want spellcasting 6. Depending on the rest of your character, you can grab a health related mentor spirit and specialization. If you do choose to focus on health spells, be sure to pick Increase X spells especially ones for mental attributes. They're one of the few ways to increase mental attributes. Increase Charisma for example will make faces love you long time.
There are some details to keep in mind. Always remember to first aid first and then cast heal. Not only does it reduce drain, it's the only way to stack first aid and heal on the same target (SR4A 252). First Aid can heal stun while Heal can't. Also First Aid works on drains and Heal can't. Both work on Fade however.
If you want to do both, I would suggest an augmented logic based mage with cerebral boosters. You'll have a good logic score for both drain and skill tests. As long as you keep your magic at 4, you'll be good caster.
2.) High Edge is going to be expensive in karmagen. However, you might be able to afford a decent edge score in the 4-5 range. I don't know. However, I recommend against "well it's an one shot burning edge is fine!" since that's extremely metagamey. That's a personal taste thing so you can disregard it.
3.) What do you mean by sense everything? You can use detection spells or have a really good perception check and sensors. With a decent Intuition and perception skill, you can get a good check. With vision and audio enhancement, you get +3 to your most commonly used senses. The attention coprocessor is cheap and essence friendly, and gives you +3 dice to perception tests. With Intuition 4, Perception 4, and gear, you can easily get a 14 perception dice pool.
For sensors, the ultrawideband radar from Arsenal lets you see though walls. Spy Games has some useful sensors as well.
