Are there note patterns on each string instrument that can not be played?

I'm going to disagree with a lot of the existing answers: On a string instrument, there are some note patterns that can't be played - at least not neatly or quickly.

Double stops - playing two notes at once - have been mentioned;

• It's not possible to simultaneously play more than one note on the same string (although you can usually use different strings unless both are too low to be on any but the lowest string)
• It's not possible to simultaneously play two notes that are too far apart for a hand to reach on adjacent strings - a sensible maximum is probably an octave (violinists might manage more with difficulty, violists won't if they're playing the size of instrument you'd expect for their hand size, cellists would like less and need advanced techniques for an octave)
• It's not possible to simultaneously play two notes on strings that are not adjacent (at least playing both with the bow)

The restrictions on being too far apart also apply to playing nearly simultaneously, although in isolation they are probably possible but messy - e.g. on a viola going directly from a note on the C string to one on the A string without hitting the intervening strings is not easy if you have a fraction of a second to do it in, even for a professional.

Where this reaches "definitely impossible" is going back and forth quickly - e.g. alternating between two notes. To do so quickly a player will either need to be on a single string and lifting and replacing one finger very fast to move between notes (easier with some fingers than others, and restricted to the reach of the hand without shifting position) or be switching strings with the bow, in which case intervening strings aren't going to be possible (and again shifting hand position between the notes constantly isn't going to work).

Yes. And the other way is also true: some sequences will sound better if played in only one string (and if you’re aware of this, better write to show this intention). Some keys are harder to stay in tune than others too, you should be cautious when writing in F minor for example. Knowing who you’re writing for usually solve most problems.

There’s not a recipe on this, but you can consult a fingering chart to visualize difficult or impossible fingerings. Also, a textbook on orchestration will help you a lot - check Samuel Adler’s book.

Generally speaking any single note (within the range of the instrument) can follow any other single note. If it is an awkward combination it just requires more skill.

The problems come with chords. There two notes which have to be played at the same time pretty much have to be playable on adjacent strings. They can't be played on the same string (e.g. G and A on the G string) and if they are far apart then they have to be reachable within the maximum stretch that the musician can manage between first and fourth finger for violinists/violists and thumb and fourth finger for cellists and bassists.

For a violinist octaves are perfectly fine, e.g. A on the G string, played with the first finger, and A on the D string, played with the fourth finger. You could extend this up to a tenth, e.g. A on the G string C# on the D string, but anything more is going to be beyond the capability of anybody who doesn't have large hands and long fingers.

On larger instruments, e.g. cello, this becomes even more difficult although a cellist might still manage this by using the thumb instead of the first finger.

Because string instruments have multiple strings and fingering options they generally don’t have as many problems as some woodwind and brass instruments can have with certain sequences, trills and jumps. My suggestion is to read a bit about how string instruments work and how to write for them in an orchestration book. Be aware of the ranges of the different sections and write within those ranges. Write things that are musical to your ear and they will more than likely be playable. If you have any doubts or concerns about a particular phrase or passage consult with one of the string players in your school.