Viharas were for the purpose of living, Chaityas were assemblies for the purpose of discussions. Further, Chaityas were with Stupas, Viharas did not have stupas.
Seven major pillar edicts have been found at Lauriya-Araraj, Lauriya Nandangarh, Rampurva, Nigali Sagar, Sarnath, Topra, and Meerut. Ashokan pillars probably signify the axis of the world (axis mundi) that separated heaven and earth.
Greeks, Kushanas & Shakas had entered India through the North West side. Accepting the local religion would provide them the easiest way to achieve legitimacy of their rule. Since Buddhism was prevalent in the area at that time and it was provided easiest entry (Hinduism had st5rict caste based structure) so, they embraces Buddhism.