Ok, so this is all just in my opinion:
Honestly, just don't swing really at all early on in the game/set. Only the safest of safe swings in the early going, not even really dtilt unless you get them near the ledge. If he doesn't want to play the game of chicken, it works into your favor as you gather data to swing more intelligently and often as it goes on. This applies to a lot of characters, but especially Falcon because of his speed to punish you, but lack of a projectile to safely attack you with. Basically, you don't want to flat out approach him, but you don't want to play too defensively either or he will likely crack your defenses and blow your face apart with his combo game. You have to find the right mix so as to keep them guessing and afraid. If you are getting hits but not converting well, it is likely you are playing too defensively while they are attacking intelligently. At that point, you need to either start reading their approach or actively denying their approach. Denying is much safe than reading, and really what you want to do. I don't just mean stuffing, his moves, I mean positioning such that trades favor you, moving ambiguously so that they feel the need to overshoot, etc.
An example would be DDing at various lengths and ranges, forcing them to overshoot their nair to cover more ground. You then dash forward a bit once you see the jump and shield. They hit their nair high on your shield so you WD towards them which makes them dash away towards the ledge. At this point you get an actually good mixup with favorable odds. They have to make a quick decision on a defensive action since they cannot dash back and you can space an attack to beat any of their attacks. At this point, I would likely fair if I had no read on what they were going to do. Immediate fair is the safest option imo because it is similar or better reward to grabbing depending on percent, but doesn't outright lose to anything Falcon can do. Dtilt loses to jump and grab loses to spotdodge, both potentially reversals . If they jump, the fair hits them, they go into a horrible position from which they will likely die. If they shield it then they are in a much less bad position, but still pretty bad considering Falon's OOS options, especially without WD back. If they spotdodge, they are pretty much frozen into another mixup.
To actually answer the post in a succinct way:
Approaches: don't approach if possible. feign approaches to threaten and force reactions. If you aren't getting the reactions you want, press a bit harder with relatively safe zoning. if that isn't getting it done, add a layer of relatively low-risk approaches like dtilt or simply dash forward. don't approach through the air unless it is very low commitment or very high reward (they are in a bad spot or you have a read) preferably both.
Platform Punishes: I rarely play against players who can shield drop, so I won't really say anything on this. Just know your percents for CCing and have the stuff in the kadano thread about uair through platforms down.
Beating nair: CC if you can, make him hit high on shield if you can, make him over/undershoot with ambiguous movement, try to trade/stuff it if it will favor you, and punish him for getting greedy to discourage its use. Really, Falcon's nair vs. Marth doesn't have very solid counter-play if they use it well, you kind of have to do a lot of little things to help mitigate it. It just has too many upsides vs. Marth to be shut down without a solid multi-dimensional strategy.
Punishing from throws: The goal should always be to get him offstage enough to set up an edgeguard. That is Falcon's most vulnerable position by far. Whatever throw goes furthest towards achieving that goal is the one you want. Juggles are cool, but I will take 30% and an edgeguard over 100% and trying to cover his landing every time. I would rather carry him off the stage with fair/nair or simply hit him off with fsmash asap than juggle, his recovery is just too free, especially on stages with low/no platforms and even if you don't kill, you will likely rack up just as much percent through the edgeguard. Again, know your guaranteed stuff from the kadano thread and remember that a set-up edgeguard is the best punish you can get on Falcon outside of an outright kill.
