I know this is just a glimpse of how I use Shulk, but just wanted some criticism and find out what are some fundamental things I should focus on? In general I think I need to get an understanding of what's safe, what isn't safe, and how to use the Arts properly.
Here are some general things I'm picking up from your video, hopefully you find them useful.
First, avoid aerial counters in general. When airborne, Shulk cannot use his much better forward counter (hold forward after the counter activates for a faster, stronger activation), and even if you get the activation, the wonky angle Shulk travels at will often make you whiff, making counter punishable even on hit. I know it hurts to have so few landing options, but in my opinion, MALLC (with Bair, Dair, or airdodge) is a much better option to explore there.
Second, I see you doing rising Nairs and falling Fairs in these two videos - you want to reverse that. Nair has no autocancel, but little base landing lag. If you go through the whole attack animation, it will be extremely punishable, especially on shield. If you focus on landing right after the attack connects at max distance, or on crossing up your opponent with the early hitbox, it will be much safer and easier to combo with. Fair has significantly more landing lag (and, just as a note, it autocancels from full hop, though that's still not great) and thus is much less safe to approach with. Instead, I like to use retreating fairs to punish approaches. It's not perfectly safe, but it is much less punishable than landing right in front of them.
Your art management is pretty good. It looks like you know that you can change arts while in hitstun, but you're still a bit slow on it. Remember that as soon as you get hit, 4 quick B-inputs gets you from any art into jump basically as soon as you can mash it out. On those lines, something that not everyone knows is that if you are drifting when jump/speed/shield art activates, you will keep vanilla Shulk's air speed until you apply a new drift vector after the activation.
MALLC is obviously something I need to learn how to do as well, as well as implementing more pivot tilt and pivot grab (esp. with Speed art). I also need to learn which throws to use and when (down, forward, back, or up). Oh, and can't forget my OoS options (I'm very used to Samus' Up+B OoS so I should be able to pull that off with Shulk too I think?).
MALLC is useful, but don't fall into the trap of thinking it's a magic bullet. Get comfortable with switching arts during the flow of combat, and the tech will naturally put you in advantageous situations, but I'd caution against trying to drill it as a thing to use all the time. Keep in mind that only the first B input (putting you into art selection) needs to have a neutral stick input. All future inputs (cycling through arts) can be done while running, walking, drifting, jumping, etc.
And yes, Air Slash OOS is AMAZING. Learn it and love it, it's hands down one of Shulk's best kill options. Beware that, compared to Samus, you'll need to be more conscious of the direction you input it in. Its range will allow you to scoop up people from much further away, but it won't be as easy to hit people directly on top of you.