1. If the piece that you are playing is intended for F#,B,E,A "solo" tuning, then 99.9% of the time you play as if your instrument was in orchestral tuning, and it comes out sounding higher. This sometimes takes some getting used to, especially if you have perfect/very good relative pitch, or you've been working on it in one tuning and then switch to the other.
2. Yes... Maybe. It really depends on what you are playing, and how it was written. Some bass repertoire is written very much with solo tuning and 4ths in mind, and becomes awkward or not possible in other tunings. For example, if it takes advantage of open strings that you no longer have in a different tuning that can become awkward, but things like harmonics (which are pretty common especially in modern solo bass repertoire) become impossible. Some repertoire was written with a different tuning in mind such as Viennese tuning for Dittersdorf and Vanhal. This makes cadenzas that use a lot of chords and harmonics challenging, and there are plenty of additions that make adaptations or use a more "4ths friendly" cadenza. Finally, some music is just music. The tuning doesn't really factor into it all that much, it is playable in multiple tunings etc. The Bottesini concerto in B- is often played in A-, and works fine in solo or orchestral 4ths, and sits very nicely in 5ths as well.
3. Typically, solo strings share much of the characteristics of their orchestral counterparts. If you play a particular orchestral string and like them, if they are offered in solo tuning there is a very good chance you will like them. This takes a lot of the guess work out of finding solo strings for a lot of people. If you play an orchestral string that has different characteristics than what you are looking for in a solo string, then you might want to look at something different. Keep in mind if you are buying new solo strings and using them rarely, they are going to sound like new strings that have not been broken in. If that brand new Spirocore twang is something you can't stand, then either put them on and break them in for a while, find them used, or go with something else. If the strings are going to be going off and on your bass repeatedly, there are some strings that deal with that better than others. Dominants are considered pretty problematic in this regard, and their solo A's have a habit of breaking even if they haven't been on and off a few times.
3.1 Depending on the music you are playing, you might only be using the top string(s) and you can get away with leaving the bottom string(s) in orchestral tuning. Look at your music very carefully and see if you will be using a fingering that would use the other string(s) or if there are required harmonics etc. A lot of players that have extensions on their basses or 5 strings keep their lowest string(s) in orchestral tuning as extended F# strings and solo C# strings are very rare, and removing an extension and putting it back isn't realistic for many players. If you have a few notes that go down to those strings you can also learn them at the correct pitch. That can get confusing and is a bit risky in a performance setting, so it is your call.
4. Some publishing companies are sneaky. Bass music tends to be quite expensive anyway given how small the market is, so being tricked into something sucks. Some publishers release editions that have multiple parts in them, allowing for solo tuning, orchestral tuning or Viennese tuning, corresponding piano parts, and even "cheat" parts for if you want to play on a Viennese tuned bass but "read" the music as if your instrument is still in orchestral 4ths. My brain doesn't work that way, I read everything at the pitch I am playing, and then I adapt mentally if my bass is tuned in 4ths, 5ths, Viennese tuning, or something else some composition student thinks is a good idea.
If you are planning a recital/solo performance, be mindful of what else is on the program. If you have access to multiple instruments that can be in multiple tunings that is great, but if not you will have to make some choices. Most of the solo bass repertoire is available with orchestral or solo tuning piano parts for this reason, some of them are a little harder to find.