Regarding Question 1:
When you are at home with your SB, your iPhone and your SB are both talking directly to your router. Data transfer from the SB goes directly to the iPhone via the router and not out through the Internet or your ISP. With most routers, this transfer will be very, very, faast.
When you take your iPhone OUTSIDE of your house, then your SB must upload its streaming data to the Internet via your router and modem, courtesy of your Internet Service Provider. The UPLOAD speed of your Internet service must be fast enough to support this data stream -- especially so if you are attempting to stream your TV programs in HD.
Also, when outside your house, your iPhone must be connected to the Internet to download the video stream at sufficiently fast speed. If your iPhone is not connected at that time of viewing to WiFi with that Wifi's ISP providing sufficiently fast DOWNLOAD speeds, you will get stuttering and stalling. And if you are attempting to watch on your iPhone from a 2G or 3G phone connection, instead of from WiFi, you can pretty much forget it.
Regarding Question 2:
If you only have one wire set up to provide video input to your SB, and that one wire (and only that one wire) then connected on the output side of the SB to provide video to your TV, then whatever changes channels on the input side (your iPhone or whatever) will also change the channels on the TV.
Many TVs have the capability to accept multi signal inputs, and you use the TV's remote to switch between those inputs. For example, my TV has one input source for TV signal direct from the wall, a second input for video signal from a standalone DVD player, a third input for video signal from my cable box passing through my Slingbox to the TV, and a fourth input for video signal from my standalone VHS videotape player. ONLY if the TV is set to display video input from "Source 3" will my remote computer activities with the SB affect what is displayed on the TV screen.
Yes, the SB does have a built-in Tuner, and if you provide TWO input signals to the SB, but have only ONE output signal passing through to your TV, then you can use the SB's built-in Tuner to send video channels over the Internet without changing what is displayed on the TV. There are lots of ways to set up this capabilty -- it depends on how your TV broadcast service is providing channels to your home. In my case, I put a signal splitter directly on the coax wire coming from my wall. One wire goes to my cable box, and from the cable box a set of 5 component wires (red-blue-green video and red-white audio) go to the input side of the SB, and another 5-wire component set goes from the output side of the SB to the TV. Anytime I change the channel on the cable box (whether in person with the cable box's remote or via my computer over the Slingplayer software), the channel on the TV changes and the channel set via SB over internet also changes.
The second wire from the signal splitter goes to the coax input connector on the SB, and there is NO wire on the output side. I then setup my SlingPlayer software for this "second input source" and programmed it with a plain vanilla remote control in the computer's slingplayer software. This signal input DOES use the built-in tuner in the SB to select channels and send them out via the Internet. Since I am not using the video signal in the cable box, nothing changes in the cable box or on the TV screen when I am slinging via this coax wire input.
Note, however, that I do not get to view all of the channels that my cable box can deliver. I can only see those channels which the cable company is required by federal law to transmit "in the clear", whether or not the customer has paid for a cable box. These channels typically include the local national network affiliates, any local community TV network channels, plus the odd home shopping and religious channels.
If you want to see every possible channel being sent out through the SB while at the same time giving you complete control at the local TV of every possible channel,then you will need to rent TWO cable boxes from your TV supplier -- one for the TV alone and one for the SB.