Yo DJ Math, welcome to Falcon.
Falcon is a very hard and frustrating character to play as a beginner, but with enough practice you will eventually become a hero of the people. Before answering those 4 questions, I'm gonna give you the best tip I can possible think of when playing Falcon: Master shffling! Practice shffling all the time as fast as you can. When I started, i thought that I could shffl well, because I could do 10 shffled aerials in a row, but until you can reliably 0-80 or 0-death most floaties off a grab, you cannot shffl well. You have to be able to react quickly to your opponent's DI, and follow up with a frame perfect aerial.
With that out of the way, let me try and answer your questions.
Basically you do not want get hit, and you want to hit your opponent. A lot of new Falcons spam nair, because they see top level Falcons use it a lot, but you have to know how to use it. Falcon's nair can be used very well for both defense and offense, just make sure you shffl and space it properly! If your opponent is shielding, you want hit with the aerial as low as possible, and either jab or dash away as soon as possible. Stomps are terrible on shield, but uair, bair and fair are all safe on shield as long as they're not too stale and hit as low as possible. An unstaled nair on shield will allow you to jab on the same frame as most characters can grab. Grab will beat jab, but a lot of the time, people aren't fast enough. A perfect knee on shield will even let you grab before they can.
Falcon's offense is a lot about getting people into their shield with our spaced aerials, then try and punish their out of shield option. With practice, you will be able to read if they're gonna roll, jump, grab etc. That's when you punish with a grab or a hard hitting aerial like stomp or knee.
Falcon doesn't really approach well, but most characters don't really. Falcon's best punish is a grab. On fast fallers we tech chase with grabs and hard aerials, and on floaties we use our insane reaction time and shffl skills to destroy them no matter how they DI. Comboing will be very difficult for a long time, but eventually it will become easy and you will destroy any floaty. Tech chasing can be a little more tricky when it comes to reliable punishes. I use a mix of reaction and reading. Most people will tech in place if they're afraid of getting grabbed, and most people will tech roll if they don't wanna get knee'd or stomped. Reacting to tech rolls with grabs is very easy, or at least it will be once you get some experience. Reacting to tech in place is not easy, at least not on spacies, because they can shine out of it so quickly. For specific combo pcts, look in the index thread, and watch a lot of videos.
As far as "unfavorable matchups" go, it doesn't really matter. Most people will say Fox, Falco and Sheik in any order, but it depends on how well you know the matchups and how well your opponent know how to deal with Falcon. For a Falcon who can't yet combo like a pr0, Marth, Peach and Jiggs will be very hard to play against. As you get better, you will destroy them all with 1 grab, so rest easy
Be sure to check out both The Falcon Trials and The Falcon Tech Skill Trial videos.
Good luck!