Is there a reason we say Schweizer Schauspielerin as opposed to schweize Schauspielerin (note lower case)? I believe this has to do with adjectives based on cities always being capitalized and ending with -r (e.g. Berliner Mauer). But why is a country-based adjective behaving that way?
The origin of the name "Schweiz" is indeed the name of the town "Schwyz" and the canton with the same name.
In the 14th century the Swiss people were actually called "Eidgenossen". After a battle (Schlacht bei Sempach) where soldiers from the canton "Schwyz" had an important part, a chronicler first called all "Eidgenossen" as "Schwyzer", short for "Schweizer Eidgenossen". And in the following centuries this name pushed through.
And that's why we have today
This rule actually refers to "geographical names" rather than just cities - the capitalisation of "Schweizer" thus just follows the rules :)
Von geografischen Namen abgeleitete Wörter auf -er schreibt man immer groß <§ 61>.
das Ulmer Münster
eine Kölner Firma
die Schweizer Uhrenindustrie
die Wiener Kaffeehäuser
Edit: I apparently misunderstood the question at first. If it is indeed primarily concerned with why the "-er" derivation isn't restricted to cities' names, I'm afraid there's probably no firm rule. You'll just have to memorise the rather few instances.
Speculation: There might be a tendency for this derivation in the alpine regions:
Tirol -> Tiroler Berge
Allgäu -> Allgäuer Kässpätzle
Vorarlberg -> Vorarlberger Skilehrer
Schwarzwald -> Schwarzwälder Kuckucksuhren
but also: Thüringen -> Thüringer Rostbratwurst (although "thüringisch" is used for everything else besides these sausages)
Today, Schweizer is considered an adjective. However it does not possess any declension which points to its actual origin being a genitive plural of the noun Schweizer. This also is the reason why it is still capitalized. Thus Schweizer Schauspielerin actually meant "(der Schweizer) Schauspielerin" thus "Schauspielerin von den Schweizern".
Knowing this, it is helpful for writing fluid German to avoid ambiguities that may reactivate the original genitive. Thus writing "... der Schweizer Schauspielerin" is harder to read, as "der" may be genitive plural or genitive/dativ singular.
Within Switzerland, the adjective schweizerisch is preferred to Schweizer. The official name of Switzerland is Schweizerische Eidgenossenschaft. Similarly Schweizerische Normen-Vereinigung and not the Schweizer one &ct.
Sorry, not enough rep to comment. I think you can say "deutsche" und "englische" because it already ends with a "sch". "Schweiz" does not end with "sch" so it's not possible.
You could use "schweizerische Schauspielerin" which seems to be also correct. https://www.duden.de/rechtschreibung/schweizerisch