Does having the Eurail pass allow you to get tickets/reservations as early as buying them separately?
As far as I know they are under the same rules as what the train operator allows. We had to wait for example on when we could book our seat reservations for the Brussels to Ghent, Belgium segment as the train operator only opened purchasing of tickets a month in advance; we had bought a combo ticket with Eurostar that allowed us on the same day of travel an additional ticket anywhere in Belgium after Brussels but couldn't do the seat reservations until the Belgium train operator opened up that ability.
I think they try to warn people about train reservation with the Eurail because some people might not realize that seat reservations are a second step that costs extra (sometimes unavoidable cost if the train operator/route requires reservations).
We have specific days we want to travel and can't afford to miss the train or have it full.
I know exactly how you feel because I felt the same way but unfortunately that is how train travel works. The ones I can say have bad reputations are Deutsche Bahn (Germany's system) though we did not experience a delay that impacted us and Eurostar for rail strikes though it can hit other train operators as well.
For Europe rail strikes are legal for most countries and you need to make sure you keep up to date on when the will strike. For example, and this isn't meant to scare you just inform you, there have been strikes that cut off England from the rest of Europe leaving passengers without a train route there. Usually these things are planned in advance. However, airports (either an airline employee, baggage handling, or security employees) also have legal strikes allowed (Heathrow frequently is part of it) so unlike the U.S. where these things are considered illegal there they are legal and frequent enough. Be it airline or train its hard to avoid this being a possibility unfortunately.
Just keep an eye out on the news and every now and then search.
As far as full that can happen. First class is treated different than second class as in I don't know that you can do standing there. However, generally if the seats are full for second class you *should* be able to stand in certain areas of the car however that's without me knowing how Scotland handles it. We had to stand on our 40 min journey from Copenhagen, Denmark to Malmo, Sweden as first class was sold out but we didn't have our luggage luckily as we were touring Copenhagen that day with our hotel being in Malmo. However, quite a few others did have luggage. You make due with trying to either use the luggage racks (which there was a central one for the train from Malmo to Copenhagen) or keeping it in your hands/control.
I do agree with the PP to try and keep luggage light. We traveled with carry on luggage (slightly smaller than standard U.S. size) and a personal item (that was a backpack) and did laundry in our hotel on the days we could.
I guess I just don't understand how train travel works as I can't imagine showing up to the airport without a ticket.
I know we felt the same

So for Europe like I mentioned we pre-purchased basically all of our tickets but we just got back from Japan about 2 weeks ago where we also traveled around central Japan via train and there we only pre-purchased Shinkansen (bullet trains) and it actually made me more nervous there without having things pre-purchased.
My personal advice is to try and book trains that allow a decent transfer. Decent in airline travel is a whole 'nother definition but IME decent for train travel in Europe is at least 15mins. We did have some incredibly tight connections (6-8 mins) and that was not my cup of tea. One connection we barely made it as in we made it to the platform right as the train came and it left quickly enough after. There can be walking in between where your train drops you off and where your next train takes off. Generally if you miss your train it's not the end of the world as another will come along there just may be some time in between that.