I'll start with the most important part: Find a good job first. As is reinforced on the MMM blogs so many times, you really want to find a good job first, then perhaps even rent for a year while you make sure you like both the job and the location. "Phoenix" is a huge down comprised of 20 or so cities and stretching some 75 miles from San Tan Valley to Sun City West. You don't want to just buy a nice home in Surprise then end up working at the Intel plant by Sun Lakes, for example.
Some people get it in their minds that renting is always bad, but I personally disagree. If you're moving to a new city and taking a new job, I would suggest renting first while you decide if the location and job suit you. Renting actually helps keep you mentally freed up to look anywhere in the valley, rather than getting locked into the mindset of "Oh, I can't apply for my dream job because it's too far from the house I just purchased." That's just me though - I am a huge proponent of giving yourself options, and choosing to rent gives you a lot of options. You can weigh that against the other option of deciding how long to live with your parents.
As for specifics, I made the following somewhat humorous map maybe a year ago for a different forum where somebody asked the same question. Basically I answered with "You can live anywhere outside the NO ZONE". Of course, several guys jumped in to say that I made the no zone too big, that it isn't all bad (especially the further north you get, as long as you stay clear of the I-17 corridor) but as a broad generalization, that's the short version of what's nice and what isn't.
Now it's worth pointing out that yes, there are some areas of Peoria, Glendale, and Phoenix that aren't half bad, just as there are some cruddy neighborhoods in Scottsdale, Tempe, Chandler and Mesa. It's more about the averages though. The average within the no zone is not so great in terms of crime and safety. The average outsize the no zone is pretty decent. Gilbert was actually ranked the 2nd safest city in America last year, behind Irvine, California. There's maybe 8 square miles of Gilbert I'd rather not live in, and 30 square miles I would love to live in. Only problem is that it's way at the southeast edge of the valley and statistically likely to be far from your job. Hence, my first suggestion to find a job first and give it some time to make sure the job is right for you before putting down roots and buying a home.
As for the rest of your questions:
2) Yes, I suggest having at least one car, but you may as well get a bike and trailer too. I have been doing my grocery shopping by bicycle a lot in the last 6 months and aside from late June through the middle of September when it's only practical to hit the stores at 6am before it's ridiculously hot, it otherwise isn't bad. If you're willing to shop at dawn, then honestly I think you could do grocery shopping (most stores are open by 6-7am) from September through June. Most of June, July, and August it may already be 80-85 degrees and with monsoon humidity even at 6am, unfortunately. The area I'm in, I have access to 4 grocery stores within 2 miles. Pick a good area and you'll be fine.
3) A scooter? Maybe, but actually the biggest problem is that Phoenix is largely a car city. Maybe scooters are common in Taiwan so people are used to looking out for them, but scooters and motorcycles are basically uncommon here, even though Phoenix is considered a popular town to own a motorcycle in. Guys paint the streets with their blood all the time. Statistically you are 35 times more likely to die on a motorcycle than in a car. A decent scooter will run you at least $2k and a halfway decent car with a little life left in it should "only" be $4k or maybe $6k if you splurge. I would do it for safety alone. Fountain Hills is the only place I would even begin to consider owning a scooter, and only if I stayed within FH. (The town-wide speed limit is 35mph)
4) All of the major carriers as well as a number of small ones are totally fine. Phoenix-Scottsdale-Mesa is the 14th largest metro area in the USA. Everything works here.
5) I would look for newer build homes, like year 2000 and later. Energy efficient homes were only just starting to really get popular in the late 90's and by 2000+ they were beginning to become standard. I currently live in a 1000 square foot home built in 1986. There are so many things about this house that are stupid that it makes me want to cry sometimes, but then again it is only a rental. My power bill, even with keeping the house at 80 to 83 degrees in the warm months is only about 60% of the power bill of my parents' 2300 square foot home they keep at 78-80 degrees. My house is 1986 and theirs is 2006. The advances in insulation and efficiency with newer homes makes a HUGE difference. Electricity can easily run you $2500+ a year on an older home with a crappy A/C unit, where a newer one you might only be at $1500/year.
I guess my question to you would be: what line of work are you looking for, and what salary range do you expect to receive? That could influence where you want to live and what housing areas are affordable while still holding back a healthy percentage for savings every year.
