Hindi: prayatn, yatn

Are these two words prayatn and yatn same in meaning? Chaturvedi provides the same primary definitions for both, which makes their use a bit redundant to me since they have the same base.

प्रयत्न prayatn (nm) effort, endeavour; attempt; (in Phonetics) manner of articulation; -लाघव economy of effort; ~वान्/शील assiduous, diligent, one who makes efforts; hence ~शीलता (nf); —करना to try, to make an effort; to attempt/endeavour.

यत्न yatn (nm) effort, endeavour; attempt; care; ~पूर्वक with effort/attempt; with care; ~वती feminine form of ~वान; ~वान/शील making effort, effortive.

Yes, both are synonymous.

Hindi word reference: prayatn, yatn

Hindi: जब तक...तब तक + नहीं

How would you translate this sentence?

जब तक तुम काम ख्तम नहीं करते हो तब तक मैं यहा बैठी रहूगी।

"Until you finish the work, I shall remain sitting here."

Hindi word reference: जब तक, तब तक, नहीं 

Hindi: मंगलयान maNgalyaan

From a BBC Hindi article I know that Mars is maNgal मंगल but what does maNgalyaan मंगलयान mean? How it is constructed?

"Mangalyaan" literally stands for "Mars Craft" and it means "a spacecraft for Mars" or more practically related to India's Mars mission, it stands for "Mars Orbiter Spacecraft"

Hindi word reference: मंगलयान maNgalyaan

Hindi: might as well

What would be the most natural translation of this phrase "might as well"?

I think depending on the context, there may be more than one way to say it in Hindi. We can use, "bhi" or "hi" to mean the same or in some other ways too. For example:

"Since you have to wait, you might as well sit down and relax."

"Tumhe pratiksha karni hi hai, to tum  baith bhi sakte ho aaraam se." (तुम्हें प्रतीक्षा करनी ही है तो तुम तुम आराम से बैठ भी सकते हो.)

 OR

"Jab tumhe pratiksha karni hi hai to kyon na baith jao, thoda aaraam kar lo?" 

OR

"Chunki tumke pratiksha karni hai, achchha hoga yadi tum baith hi jaao"

Word reference: might as well, hindi

Hindi: windy

How would you say "windy weather" in Hindi?

One can say, "hawadaar (हवादार मौसम) mausam", or "tej hawaon wala (तेज़ हवाओं वाला मौसम) mausam".

वातिक is also one word for the same. वातिक काल (windy time/weather)

तूफानी मौसम can also mean the same, though it means "stormy weather" to be exact.

Word reference: windy, hawadaar, havadaar, हवादार, वातिक, तूफानी

Hindi: samarasataa समरसता

What does samarasataa (समरसता) mean in Hindi?

It means "harmony".

For other words, i.e. synonyms, please check out: http://www.shabdkosh.com/hi/translate?e=harmony&l=hi

Word reference: samarasataa, समरसता

Hindi: bhaagiidaar भागीदार and bhaagiidaarii भागीदारी

I want to know the meaning of Hindi words bhaagiidaar भागीदार and bhaagiidaarii भागीदारी.

bhaagiidaar / भागीदार: Partner or participant
bhaagiidaarii /भागीदारी: Partnership or participation

Synonyms of भागीदारी are the following:

 भागिता
 साझा
 साझेदारी

All these three mean "partnership", while only भागिता can also means "participation"

Word reference: bhaagiidaar, भागीदार, bhaagiidaarii, भागीदारी, भागिता, साझा, साझेदारी

Hindi word for Vichar Dhara

Can you suggest Hindi word for 'Vichaar Dhaaraa'?

'Ideology' would be best; or even 'thought process' would be good. 

Hindi: kidhar-idhar-udhar VS kahaaN-yahaaN-vahaaN

Is the Hindi word "kidhar" same as "kahaaN"; "idhar" same as "yahaaN" and "udhar" same as "vahaaN"?

Yes, for all practical usage, both sets mean the same so are used interchangeably... 

Word reference: kidhar, idhar, udhar, kahaaN, yahaaN, vahaaN

Hindi: thaa hogaa/thii hogii (so rahaa thaa hogaa, etc)

Is it correct to write the following sentence in Hindi?

"Wah us waqt so raha thaa hogaa/thii hogii"

This use of "thaa" is not needed. It should very well be written as "Wah us waqt so raha hoga" to mean "He would be asleep at that time." or "May be he was sleeping at that time."