Capital gain

In my view, you are selling the house( on the land). So the date on which the construction was complete should be taken as date of acquisition of the capital asset.

So, STCG.

It depends on the sale deed. If you are selling only the house, then short term gain, and if you are selling both house and the land, then short term gain for the house and long term gain for the land.

I'm not sure but still it seems right

Why don’t we just take cost of house constructed as cost of improvement, so won’t the whole property be long term?

If we have taken a vacant land and constructed building on it and we are selling it consolidately and if there is long term period on land and short term period on building as in your case then for the purpose of calculating capital gains both the value of land and building is to be taken separately and long term and short term gains to be calculated thereon.

I agree with Mihir and Vishnu. My earlier answer was too conservative.

Property constructed on a land purchased earlier: In case of property is constructed on a site purchased much earlier, the question arises whether the period of holding the asset i.e., the property, should be reckoned from the date of completion of the construction of the property or from the date of acquisition of the and. The correct position is that the asset consists of two components:

(1) Land and
(2) Building.

When the property is sold, the period of holding has to be reckoned separately for the land and the building. The consideration received can also be split into two parts relating to each component. In CIT v Vimal Chand Golecha (1993) 201 ITR 442 (Raj), the land was purchased in 1962 and building was constructed thereon in the accounting years relevant to assessment years 1968-69, 1969-70 and 1970-71. The building was sold in 1970. It was held that the gains attributable to land were assessable as long-term capital gains. The gains attributed to the building were however, short-term capital gains. Similar decision was held in the cases of CIT v Lakshmi B. Menon (2003) 264 ITR 76 (Ker) and CIT v C.R. Subramanian (2000) 242 ITR 342 (Kar)].
Agreeing with the above Rajasthan High Court view, it has been held that land can be considered a separate capital asset even if a building is constructed thereon. Thus, where the land is held for more than a prescribed period, the gains arising from the sale of the land can be considered as long-term capital gains even though the building thereon, being a new construction, is held for a period less than the prescribed one [CIT v Dr. D.L. Ramachandra Rao (1999) 236 ITR 51 (Mad). Also see CIT v Citibank N.A. (2004) 260 ITR 570 (Bom)]. 

In the above cases, the burden will be on the assessee to satisfy how much of the sale proceeds should be apportioned for the land and how much of the sale proceeds pertained to the structure. [CIT v Estate of Omprakash Jhunjhunwala (2002) 254 ITR 152 (Cal)].

Source: RECENT JUDGEMENT & CONTROVERSIAL ISSUES ON CAPITAL GAINS TAX. By, C.A. Vimal C. Punmiya.