I'll admit outright, there are aspects of this that I am no SME on, as I'm a bit rusty on some rules. But I do correlate information and do research well, so let's see what I can come up with, and hopefully it's helpful.
1. In the case of the Mind Probe, it's a situational judgment call on your part, as there are no rules explicitly written for that scenario, but I'll offer what I would go by. There are two key factors I would consider every time he used the spell:
- What does the target know about magic? If they're Joe Normal, and they know magic exists and does crazy stuff, then they may have no idea at all they need to be touched for the spell to work. If they are savvy, not even necessarily awakened, but someone who is security conscious and could have done their own research, or been briefed or clued in, then they could be assumed to know it's a touch based spell and the player could be outright hooped for hiding it. If they're Awakened, however, this does not mean they know how the spell itself works (there are all manner of kinds of caster, and some just pull spells out of their ass without knowing anything beyond their own magic). If they can be assumed to be clueless about the spell itself, then it comes down to factor two;
- How subtle is the mage being? If he's just walking up to the guy and doing the old pickpocket 'oops' when no one else is around, then he's going to have a hard sell ahead of him. If he is savvy, and make contact in a crowded area, say on a busy sidewalk, then he may even fool a paranoid caster (I'd call for something stealth based there, Palming likely). If the mark is suspicious, then a Con check is entirely appropriate, however said check should be modified by the magic knowledge of the mark, and again how obvious the player is being (talking to the mark, especially preemptively, would make the Con check pretty damned hard).
2. I'm going to call shenanigans on your player's part here. Overcasting does Physical damage, clearly, and while a medkit does a wonder with First Aid, the healing rules (SRA p. 252) explicitly state that healing physical takes a day of rest. A mage can't simply cast until his eyes bleed, heal up, and cast right away. Until he's rested, he is still hurting in all manner of ways, he is just stable and feels like everything is mellow and fine (he's probably drugged to the gills). Also keep in mind that the medkit needs to be refilled often, and that's not free.
3. Again it's a matter of nothing being explicitly stated, so it's a judgment call. I would say that if the intended mark has no idea that he is talking to the same person (calling for the appropriate Perception check to see through a disguise), then the First Impression bonus would stand. I'm all for rewarding smart runners, and disguises, when used right, are smart.
4. Sub-vocalization is silent. There are actually
real world versions out there now. Essentially you train yourself to send signals to your vocal cords, without actually speaking, and the mic picks up the signals and translates them to speech. It's not your voice they'll hear on the other end of the line, but no one around you will hear a thing.
5. Tailing a car (in reality) is similar to tailing a person on foot. You keep just the right distance, match their speed, obfuscate your presence by keeping other vehicles between you and the target if possible, etc. As such, if you are trying to avoid the eyes of passengers, I would call for a Shadowing roll from the driver of the tailing vehicle.
The caveat, however, is that the driver of a vehicle by RAW always uses Sensors in the stead of his Perception skill. Sensors do include visual, but they pick up on a vehicles Signature. The judgment then comes in play again, and you can either call for an Intuition check from the driver (he knows the vehicle is there, but does he find it suspicious), or, and this is stretching it a bit, you could allow the tailing driver to modify his vehicles Signature with hits from his Shadowing check.
6. If I'm not mistaken, you're looking for Loyalty (how much a Contact likes the player), not Connection (how much a Contact can do for the player). I would say this comes down to roleplay entirely. Loyalty at build represents history with a Contact. That means time spent grooming them, and maintaining. If a player wants to increase how much a contact likes/trusts them they need to find out through roleplay (conversation directly, asking around, just observing things) what needs to be done to ingratiate themselves with a Contact. Maybe that's as simple as realizing the Contact is a vegan, and changing the venue you meet at from a McGreasy's Steak Hut to something with more options on the menu, or maybe it involves a run, or an entire series of runs. Also, if they do good for the Contact (especially a Fixer) over time, I would consider just bumping the Loyalty rating as a reward along with Karma and other things.
If you actually mean the Connection, well that is largely a behind the scenes thing and GM prerogative. The runners could conceivably do things (runs, or just make the right calls to other contacts), to help their contact move up in the world. Again, this could simply happen over time (you work with the same Johnson/Fixer, their bosses are impressed, they get promoted).
I hope this isn't too rambly, and it actually helps.