Most composers of the 17th and 18th centuries "worked" as patrons of various noble or royal courts. They would not have gone on tour as we currently understand the concept -- their performances were solely for the upper-classes, not the masses. Beethoven, for instance, spent most of his career as the patron of various Austrian high nobility.

The first musical composer who has widespread internatioanl popularity and regularly spent periods traveling and performing, was probably Franz Liszt. He actually spent most of the 1840s and 1850s touring across what it today Germany, Austria, and Italy, traveling from city to city and playing the piano. He was also a genuine pop-cultural phenomenon, with the term "Lisztomania" being used to describe the intense fan frenzy that surrounded him. Basically imagine a cross between Elvis and Liberace and you'll have the right idea.

Hector Berlioz did a few times. In 1841-2 he gave 15 concerts across what is now Germany in cities such as Berlin, Dresden, Stuttgart, Mannheim, Weimar, Leipzig, Braunschweig, Hamburg, Hannover, and Darmstadt. In each city, he would rehearse the local orchestra and chorus and assemble a program of his works that he thought the locals would enjoy. He also complained bitterly in letters to his friends about which sections of each orchestra or chorus was not up to the task. Additionally, Berlioz did not speak German so translation was sometimes an issue (Berlioz evidently had not learned from his marriage to an English Shakespearean actress--she spoke no French and he spoke no English).

An interesting feature of his "tour" is that there was not yet a rail network connecting the major German cities, so Berlioz primarily traveled by coach. He had to pay for his music (orchestral and choral parts) to be shipped by post between cities because altogether it weighed 500 pounds.

In 1845 he did another tour, this time conducting concerts of his work in Vienna, Buda and Pest, Prague, and Breslau. And in 1847 he went to Russia and did concerts in St. Petersburg and Moscow, then gave concerts in Riga and Berlin on his way back to Paris.

Source: The Memoirs of Hector Berlioz, trans/ed. by David Cairns.

That is hard to answer, because it is not specific.

If you mean classical composers, as in composers of the classical period in music (around 1750-1820), it is a different answer than that for the "classical" composers of the romantic period or baroque.

First, you need to understand that a composers "job" was primarily to write music. It's not like a modern band in pop music, where you write a song and then go on a tour to perform it, hoping you will make money. Composers only... composed. Of course, some were virtuosos and did perform (Bach was knows throughout his life primarily as an organ virtuoso), especially in the later romantic periods (19th century).

In general, a composer only composed music, and sometimes if he was also a virtuoso on his instrument, he earned money through playing/touring. However, that is exclusively from the Romantic period onwards. It is the job of the performer, conductor/orchestra to reproduce the composition as faithfully as they can, while still giving their own personal touch to the piece.

Now, to your question.

In baroque the composers didn't do tours or concerts as you know it today. Composers sticked to various royal courts and they had patrons who supported them. Bach basically stayed in a radius of a few hundred kilometers for all of his life.

For the classical period

Mozart traveled most of his early life as a child prodigy, giving concerts with his family and earning some money that way. But traveling musicians were pretty common even before the classical period. Giving out full tours was not. Giving out concerts for money actually became pretty popular in those days, because more citizens had more wealth than before and it was more substainable for the composers, but it wasn't until later that performing concerts throughout Europe became really popular.

For the romantic period and later periods:

It became much more popular to make tours as both a composer of your pieces and a virtuoso. You could get rich and become really famous throughout Europe that way.

Nicollo Paganini was probably one of the first composers to do this. He was a famous violinists, and as the violin developed as an instrument, so did the technique players used. He was one of the main developers of violin technique that violinists still use today. He was very well known for his virtuoso style, and as a bonus it was easy for him to take his violin with him all around Europe, and a lot of his pieces he could played by himself, as they didn't require accompaniment.

Another example, and probably the most famous one is Franz Lizst, who was actually a student of Paganini and wanted to become a virtouso on a piano like Paganini was on the violin. Liszt was probably the first rockstar ever, women fought over him, he got extremely rich, and people just loved him and his music. There is even a term for it: Lisztomania.

To answer directly:

When someone like Beethoven completes, say, Beethoven's 9th, would he go to major European cities to play their Opera Houses? If I lived in Paris and wanted to see Beethoven live, did I have to go to Vienna?

Yes, you would almost certainly have to go to Vienna, or any other big city that had an orchestra and musicians of good quality (there weren't that many - most were in France, England, Italy and the German speaking parts of Europe) .

Maybe it's weird to think about it now, but back in those days, it was pretty hard to find a really good musician, especially an orchestra of really good musicians. The Mannheim orchestra is a good example for the classical period.

If they didn't tour, would they be well known in other countries?

The composers were still pretty well known outside of their place of residence, because their scores were available elsewhere, and other people reproduced their music.