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Do there exist spherical triangles which are not isoceles but are the union of a finite collection of (two or more) congruent triangles with pairwise disjoint (and non-empty) interiors?

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Yes. Take any equilateral triangle and divide it into six congruent triangles barycentrically. The union of three of those parts is a non-isosceles right triangle.

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(unless that equilateral triangle was right-angled) –  Tom Goodwillie Jul 17 '10 at 19:43
Thanks for your very neat answer. I had tried to decompose many non-isoleces spherical triangles without sucess until I began to think that the answer to my question might be negative. But I did not think of gluing some congruent triangles together to see if I could form a non-isoceles triangle out of them. –  Garabed Gulbenkian Jul 19 '10 at 14:52
So, can you decompose a big triangle into several congruent little triangles in such a way that none of the triangles (big or little) are isosceles and none of them have any right angles? (I don't know the answer.) –  Tom Goodwillie Jul 19 '10 at 15:28
I wonder whether the problem would become any easier if you imposed the two restrictions of not being isoceles and of not containing any right angles only on the BIG triangle? –  Garabed Gulbenkian Jul 20 '10 at 17:31
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