Okay, let's see...
You're trying to grab too much. It's a good idea to grab, but most of the time you're just standing there and hoping he'll run into the grab.
Falcon has no business staying up on a platform. Especially not with Fox under him. Get outta there ASAP. You don't have to drop through, just running off the side works too.
Dash dancing when you techchase helps a lot. I'm really not sure why, as it doesn't seem like it would, but it does.
You're messing up some of your L-cancels. Remember, you don't have to hold the shield button, just tap it.
1:23 I'm not sure what you were doing there with the shield... trying to wavedash or something? Anyway, you can actually Raptor Boost at him and if you hit the tree before you fall over, there's no lag, even if you miss. It's pretty cool, I guess.
1:26 You could've charged a dsmash. No where he really could have teched to avoid it.
I think you're messing up a lot of your wavedashes... might want to work on that.
If you could ledgehop nairs, there were a few times where you should've. Jumping off the ledge is pretty laggy. If you can't ledgehop nairs, you should learn.
1:45 The walljump was so unnecessary. It's cool, but he probably wouldn't have fulljumped there anyway. Avoid doing stuff like that.
Yeah, cut down on the dash attacks... speaking of dashing, you can actually foxtrot in alternating directions for a longer dashdance, which is useful for mindgames.
It doesn't seem like you're really comboing. I guess in part it's because you're shielding after your L-cancels, which bogs your fluidity down, but outside of techchase grabs, you're playing a poke war. After nairs you can uair or stomp or Raptor Boost to set up for the knee. Falcon is all about comboing into the knee and you're not doing that enough.
Learning to Gentleman would be good too.
Second match!
0:24 Good save, but try not to be in that position in the first place =P
0:34 If he bounces that far, dash dance after him. You should start out by dashing towards him, so if he rolls away, you can just keep running and grab him and if he rolls towards you, you can just turn and grab him. If he's getting up in place, shield (you can shield to cancel a dash or run) and shieldgrab him, whether he uses his get up attack or not.
0:38 You should've stomped him, or at least wavedashed/walked forward so you could hit him with the utilt. You've got the right idea though, the utilt is a pretty good edgeguard. Dtilt at those percentages could maybe set up for something else too.
0:43 Why not just doublejump to grab the ledge?
...you need to stop suiciding with the Raptor Boost. Just... don't use it unless you're going towards the middle of the stage or if you're absolutely sure it'll hit.
Don't be afraid to edgeguard aggressively. Fox's recovery is relatively low priority and you should be able to cut through it with most attacks.
Work on your DI a little. It'll help you survive a lot longer, especially as Falcon.
Against Fox, random shffl'd knees rarely work. His nair and bair will almost always beat you out. Try nairs instead. On a stage like Dreamland 64, you've got a lot of area to move, so use it. JC grabs are good and all, but they don't cover a lot of area and so they'll slow your Falcon down.
So yeah... Gentleman, shffl's, wavedashing, ledgehops and DI. That's what you should be working on. You might want to mix up the recovery too with some airdodges or whatever.
EDIT:
Ummm... so... in response to JonaDiaper.
The ftilt is okay... I personally don't find it as useful as the dtilt or the utilt. Really, it seems more like the poor man's Gentleman.
Yes you can knee into a grab, but that's only at like... less than ten percent.
And yeah, you can techchase into a stomp, but stomping as an approach tends to not work so well because of Fox's two sex kicks.