(i)Well,the circular feature of Dasein is pretty much in place in the opening pages of BT.How we go about our life -whether our practical or theoretical engagement with our world -presupposes an understanding of Being. When asked, we are at loss to to clearly articulate what such a Being means.In a footnote Heidegger cites Pascal's claim about the absurdity of having to presuppose Being in order to define its meaning.As circularity can not be entertained,the question of Being can not be but a nonstarter.This is the one of many questionable ways tradition,according to Heidegger, has dispensed with the problem of Being.
 
(ii)My understanding of Samkara is based on my reading of Radhakrishnan's  Indian Philosophy(vol-ii).I wish I had the expertise to read Vivekachudamani.But I believe,that for a beginner Radhakrishnan is a good guide.Radhakrishnan is at pains to establish the fact that Samkara is as much a realist as he is an idealist.Notice the following quotations.

Whenever he(Samkara)denies the reality of effects he qualifies his denial by some such phrase as "different from Brahman" or "different from the cause".Nowhere does he say that our life is literally a dream and our knowedge a phantasm.(p-586)

Samkara holds that we can not construe the ralation between Brahman and the world in any logical way....though the world and Brahman are not regarded as complementary elements in a whole,they are not set in absoulete antagonism.(p-586)

You will find many such quotations affirming the reality of this space-time world that invites comparision with that of Heidegger.Ofcourse there are differences.Nonetheless both,in their own way,defend a realist conceptionn of time.

(iii)What you find 'weird' in Heidegger is what many people admire in Heidegger.Heidegger's disenchantment with transcendental philosophy ultimately led him to poets,particularly to Holderlin.
 Pesonally I feel Gadamer was right when he said that all his life Heidegger was searching
 for God.This is another reason ,tempting enough,to situate him within the tradition of oriental thought. 

I have oversimpliefied the matter and wonder whether this will help you.

Cordially Yours

Bijaya