1. Can Level 4 be taught in a classroom?

The simple answer to this is yes. Teachers have been taking students from Level 3 to Level 4+ for many years. A number of institutions in countries other than the United States do this routinely. In the United States, however, there are few classrooms that purport to have this goal. There are four that do that the CDLC knows of at the moment, and most members of the CDLC are planning to develop courses in a range of languages to do this, based on the limited expertise available.
Two of the experienced programs are at the Foreign Service Institute. For more than twenty years, the FSI has taught at this level. The success of students is determined by pre-test and post-test, both tests being the ILR proficiency test. The Russian program at the FSI from 1983-1989 educated a number of Level 3-4+ students with a 100% success rate, as determined by pre-testing and post-testing. That program continues to this day. In addition, around 1990, the FSI developed a similar course in French.
The Specialized Language Training Center in Rockville, Maryland, which is a member of the CDLC, has routinely helped students to reach Level 4 and higher since 1987. Again, pre-testing and post-testing determines the success rate. The test parallels that of the ILR and is given by a former ILR tester.
A little known program that teaches students who will be working on the popularly called "Hot Line" between Moscow and Washington also prepares students at and to this level. Once again, students are pre-tested and post-tested. In this case, the test used is an internally developed instrument, specific to the needs of that program.
Middlebury College also some years ago taught Arabic to this level. However, that no longer appears to be the case. Nonetheless, CASA in Cairo does this routinely.
The more complex (and more accurate) answer to this is that it depends on the background of the student coming into the course and the duration of the course and a host of other variables that are addressed in this FAQ and elsewhere.

2. What background should a student have prior to undertaking study that has a Level 4 goal?

Experience shows that in addition to having Level 3 proficiency at the outset, a student should have considerable in-country experience. The latter could include a job in the country where the language is spoken, a professional-level internship, or some other experience where the student has had the opportunity to live with friends (or roommates) and use the language in a professional way, although language skills might have been limited (only Level 2+ or 3).

3. Is study abroad good enough for in-country experience in order to reach Level 4?

That depends on the study abroad experience itself. If the study abroad is spent in a classroom with other American students, then it is unlikely that this will be enough. Further, if the student lived in a dormitory or in a complex with other Americans, the chances are that the range of informal experiences will be an insufficient base for higher level study. (In that case, it is unlikely that the proficiency level of the learner will have reached a minimum of Level 3, either, and Level 3 is a base platform for classroom study to Level 4.)
Most learners who have reached Level 4 have not only spent considerable time abroad, but that time has had similar characteristics to the time that native speakers spend with their own language. Specifically, most acquired foreign degrees, studying in classroom with foreign peers and in isolation from other Americans. (This does not mean that earlier study abroad experiences which were quite different were not valuable; at lower levels of proficiency, traditional study-abroad programs can be very helpful and bridge the student's passage from informal proficiency to formal proficiency.)
Many learners who had reached Level 4, when queried, have stated that everything that they needed to acquire from studying abroad on traditional programs and interacting informally with native speakers was accomplished by Level 3. More of the same did not help them reach higher levels. Rather, some kind of formal language learning-on-the-job, in a foreign classroom (studying any discipline that they would study at home), or in a high-level language classroom at home-seemed to be the key to reaching the highest levels.

4. Are there alternatives to living and studying abroad?

There are alternatives to intensive living and studying abroad that seem to have worked for some learners, although nearly all Level 4 language learners stress the importance of in-country experience. In cases where in-country experience is difficult to have, some learners have become a part of an émigré community in the United States, in effect, living abroad at home. Others have held jobs that have required high levels of proficiency in which they worked alongside colleagues who were native speakers.