Answers to your questions:

1. The arrival of the Jews, chased out of Germany in the mid-14th century by the "Black Death" and by mass killings, into the expanding feudal Grand Duchy of Lithuania (GDL). With the privileges (rights) granted them by the Grand Dukes, Gediminas (1316-1430) and Vytautas Magnus (1380-1430), and later on, under the direction of the state-acknowledged Jewish Va'ad (Council) (1623-1764), they carried on an autonomous community life in the Grand Duchy for a period of 400 years. The Jews contributed a great deal to the GDL, and in time its capital city of Vilnius became a Jewish spiritual and religious center. Before 1800 CE the population of the GDL included 250,000 Jews.

The earliest Jews received Southwestern part of Lithuania.

Now it's a little bit difficult for me to answer exactly when Jews became part of Rumsiskes. I will try to clear it as quick as it would be possible in archive. According to V. Sirokomle (a Polish poet) "in 1860 Rumsiskes were full of Jews". There were Jewish schools in 1831, a synagogue in 1861, and a public house. There were more than one-hundred Jewish families before the First World War. It is believable that there were Jews in Rumsiskes in the middle of the 17th century.

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2. The Jews in Rumsiskes were merchants, traders, artisans, and many of them were occupied in floating wood by river to Russia.

A few days ago I found in the "Jewish Encyclopaedia" (Vol. 9, Saint Petersburg, 1907-1913, in Russian) that there were 308 Jews in Rumsiskes in 1847; in 1897 there were 411 Jews (1,180 inhabitants in all).

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I found some material about the Jews in Rumsiskes (from 1833 until World War II), in Hebrew, by Dov Levin's "Pinkas Hakehillot" (Jerusalem,1996).

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A few interesting sentences about people of Rumsiskes from the article "Old Rumsiskes" by A. Miskinis is in the book "Kauno marios" (Vilnius, 1981).
Pages 59 - 60: In a document of 1798 there is mentioned that merchants of Rumsiskes (two of them were very rich) were occupied in iron, wood, salt, and trade. Salt was brought from Karaliaucius (Kenigsberg). Wood was sold in Vilnius or in Prussia. Thirty-eight Jews lived there in thirteen houses.
Page 61: In 1830 thirty pupils attended school in Rumsiskes (five of them were Jewish).
Page 612: There were eighty-five Jewish families in Rumsiskes in 1852.

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About 700 Rumsiskes Jews were killed on August 29, 1941 in Pieveliu village (now territory of Ethnographical museum). I don't know how exactly they were killed, but it is possible to look for information in the archives.