Despite of the advantage having access to all Einstein’s papers on relativity collected in two volumes and translated into one language, Russian, as well as the possibility to obtain copies of all original papers without even leaving our house, we experience some difficulties in answering the question in the title.
In contrast, the overwhelming majority of GW have definite answers to the double-question in the title: “Yes” and “Yes”. This conventional view, combined with the fact that nobody observed GW yet, leads to a curious but logical conclusion, for example, see :
“A couple of years after the discovery of general relativity, EINSTEIN proved the existence of gravitational waves by linearizing the field equations of his theory. But the experimental discovery of these waves still elude physicists. This leads to the suspicion that these waves may not exist in Nature. The question then naturally arises: can we set up an alternative approach to gravitation which will theoretically rule out the existence of gravitational waves…” [underlined in ours]
Such a natural conclusion was made in 1983 and since then there is no change in experimental results.
The absence of Einstein’s direct answer to this question and even disappearance of a term “gravitational waves” from his later writings leads us to wonder: how one can possibly argue that his answer is “Yes” or “No”. In particular, the arguments in the paper , quoted above, in the case of “No” will be void and there is no need for modifications of GR based on non-observation of GW.
We think that we found a key, or rather a methodological approach, to treat this problem [to find the answer to the question formulated in the title], which is concealed in the editorial preface to the fourth volume of collection of Einstein’s papers :
Работы Эйнштейна отличаются от работ многих других физиков нашего времени тем, что он аккуратно фиксировал в них все этапы своего труда, завершившегося рождением новой физики. Каждая из статей, каждое выступление добавляло новые элементы к создающейся картине физических явлений. Собранные воедино, они, подобно кусочкам мозаики, образуют увлекательную историю физики в ее наиболее бурные годы. Именно их логическая последовательность и целостность исследований заставили выбрать такой вариант издания, в котором были выделены два направления – теория пространства и времени (тома I и II) и теория атомных и статистических явлений (том III)… логика исследований проявляет себя ярче, если статьи этих двух направлений читать, не перемешивая.
Einstein’s works differ from the works of many other physicists of our time, because he accurately recorded all stages of his work ended with the birth of new physics. Every article, every public speech added new elements to the arising picture of physical phenomena. Collected together, they are like pieces of a mosaic that form a fascinating history of physics in its most exuberant years. It is their logical consistency and integrity of the research, caused us to choose this form of edition, in which we allocated two directions – the theory of space and time (volumes I and II) and the theory of atomic and statistical phenomena (volume III)… the logic of research manifests itself brighter if the articles of these two directions are read separately, without mixing them.
Note that the Russian collection was published in 1965-1967. The choice of a different approach was made by the Einstein Papers Project  that started much later (in 1970s) and so far reached only Einstein’s papers published in 1921. With such a speed, the year of 1955 will be reached in 2060! The whole collection would be of enormous size (30 volumes ) and it would be a mixture of everything: “letters to Mistresses, Mom and a Bit of Science” .
[Our estimation of expected project completion: the first volume appeared in 1987 and today, 2012, 12 volumes are published, i.e. roughly one third in 25 years, 50 more years are needed.]
In our opinion, such a collection could be of interest for future historians, but not for physicists. On the supercover of “Ideas and Opinions by Albert Einstein” (first edition appeared in 1954), one can find a quotation from The Saturday Review: “Great thoughts of a great man, assembled for the first time into one volume”. What about assembling “into one volume” the scientific achievement of a great scientist? An analog of the Russian collection in English is a long time overdue and the only explanation of such a choice (whoever made it) is a treatment of Einstein as some historical figure. He was a great man, but he was the great scientist above all. As it is said in the editorial preface to the first volume of the Russian collection :
…работы Эйншейна ни в какой мере не стали достоянием архивов…
…by no means Einstein’s works become the property of archives… [they continue to be important for today’s science]
This approach, not taking pieces of mosaic, but considering the whole picture, is the only key to find the answer. Einstein’s search for general relativity was not short (1905-1915) and different approaches/attempts were made by him before 1915, when for the first time equations were written as we know them today and call them Einstein’s equations. There are lots of historical investigations about this (1905-1915) period. One, of course, can argue that one of his old attempts (abandoned by Einstein prior 1915 ) is correct, but it cannot be presented as Einstein final view. In 1915  GR equations were written in short “letter size” paper and more detailed exposition was published in 1916 . In the same year, 1916, paper  was published and becomes the common reference in popular and scientific literature on the proof of GW existence. For the next almost 40 years he continued to work on his the most important contribution, GR, (his own view) and he used to refer (and not once) to the words of G.E. Lessing, which are also found in the last sentence of his autobiographic sketch written a few months before death :
Das Streben nach der Wahrheit ist köstlicher als deren gesicherter Besitz.
The search for truth is more precious than its secure possession.
He searched for truth the whole life, and taking some paper in the beginning of his search, without a comparison with his later writing on this subject, is the same as taking pieces of a mosaic without looking what part of picture they are, or perhaps, it is just a part of an initial sketch.
In 1916 in paper , Einstein considered a solution of the linearized equations of GR. After linearization is performed, one familiar with the Maxwell theory, can by analogy find a wave solution. However Einstein was always interested in principal questions, and solving similar to Maxwell equations can hardly be called principal. The validity of linearization for non-linear, generally covariant equations of GR is indeed a principal question. Let us read his 1916 paper , find what he thought about linearization and compare with his later writing about it.
To be continued
 M.N. Mahanta, Lettere al Nuovo Cimento, 38 (1983), 302-304
 Альберт Эйнштейн, Собрание научных трудов в четырех томах, Издательство “Наука”, Москва, т. 4 (1967)
 Альберт Эйнштейн, Собрание научных трудов в четырех томах, Издательство “Наука”, Москва, т. 1 (1965)
A brief information about the editors of Альберт Эйнштейн, Собрание научных трудов в четырех томах, Издательство “Наука”, Москва, тома 1-4:
I.E. Tamm (the Nobel Prize winner): http://en.wikipedia.org/wiki/Igor_Tamm
Y.A. Smorodinsky (a graduate student, college, and coauthor of L.D. Landau):
(the author of a very popular in Russia biography of Einstein, that was also translated into French: B. Kouznetsov “Einstein – sa vie, sa pensée, ses théories”
(our bilingual (Einglish-French) Canadian frends found it very interesting)
Note that the translation of scientific papers (and majority of non-scientific) were performed by Russian theoretical physists.
 A. Einstein, Sitzungsberichte der Preussischen Akademie der Wissenschaften zu Berlin, 1915, 48, 2 844-847
 A. Einstein, Sitzungsberichte der Preussischen Akademie der Wissenschaften zu Berlin, 1916, 688-696
 A.Einstein, Annalen der Physik, 1916, 49, 769-822
 Albert Einstein, “Erinnerungen-Souvenirs”, Schweizerische Hochschulzeitung 28(Sonderheft) (1955), pp. 145-148, pp. 151-153
(http://www.ethistory.ethz.ch/texte/1955Einstein_SHZ.pdf ); Reprinted (without first two paragraphs) as, “Autobiographische Skizze” in Seelig Carl, Helle Zeit – Dunkle Zeit. In memorium Albert Einstein, 1956, Zürich: Branschweig: Friedr. Vieweg Sohn/Europa pp. 9-17 http://www.philoscience.unibe.ch/documents/kursarchiv/WS99/Skizze.pdf
(we were not able to find, unlike German, a freely accesable English translation)