I think you are focusing too much on after you get hit. It's good to know how to DI and stuff, but there still are combos in this game. Sometimes you can't do anything but DI weirdly and hope they don't follow you correctly. What you are having trouble with is probably beating the spacies in the neutral game. If they feel no threat by being near you, then they will just jump all over you and eventually your defense will crack and you will start getting combod. Similarly, if your attacks (run in and JC grab) are too obvious or poorly timed, they will easily defend and then get hits on you.2) I get repeatedly juggled by the spacies (whether its Falco with his dair shine or fox with his up tilts, back airs, up grabs, and so on). I try to DL but this hasn't been succeeding. Is there a certain way to specifically DL to escape the pressure / any other alternatives I should focus on?
3) This is more of a note in regards to my mentality in the matches I play that I require assistance on. Whenever I play Spacies, I feel as though I get outspeeded (like when I go for my JC grabs that usually do not land). From there, I feel threatened to shield (causing me to be juggled by their shine combos and what not), or shield too much (causing them to shine-grab me). How do I overcome this mentality and begin punishing them brutally as Sheik? (More of a add-on to question 2 I'd say).
You need to know how to beat their attacks - not through shielding, but through evading their attacks - and how to attack safely. You need to know when you should throw out that grab and when you should just run around a bit more. Maybe try to throw some needles (I am assuming you play Shiek). If you jump and throw them before you land, you do not have any lag for throwing them. I don't have too much to say on this part because I am not a Shiek expert.
The important message I am trying to convey is that you should not look at it as "what do I do when I am stuck in my shield getting pressured by a spacie" and look at it more as "how can I avoid getting stuck in my shield". Probably the biggest thing you will find is that if you are too far away when they land to hit you with their shine, then you pretty much have a free shield grab. Learning to set up this scenario will get you far against spacies, since with Shiek, you can tech chase grab them for a long time. You can set up this scenario by moving around a lot, and then if they approach at the wrong time (a little too far to do a proper aerial->shine), then you get to punish them for it.
I hope this helps.
EDIT:
I wanted to add a sidenote about them being faster than you. This is the result of them being free to move. They are free to move if they are not threatened. You can threaten them with very powerful defensive moves (dash dance grabbing their misses etc) and by attacking only safely (throwing needles when you can, jabbing their shield and then backing off, etc).