I use GuitarTuna app for tuning the guitar,I am able to get standard E tuning with the app, For other drop Tuning like Standard C or D, putting capo on 2nd for D standard or capo on 4th fret C Standard, then match the notes to standard tuning with the help of app, and then at last when we remove capo, will it give us D Standard or C standard tuning for sure?
In theory it might work like you describe, but tuning through a capo feels like a terribly inefficient kludge. The capo will try to prevent the strings from moving under it easily, that's for sure. If the tuner app doesn't directly support other tunings, you could use a chromatic tuner and figure out what the correct notes are for your target tuning. For example, if it used to be E and you want it to be a whole tone lower, it becomes ... D! Set the chromatic tuner's target to D and tune the string. Do the same for all strings, it's not rocket science. :)
It may - or may not work. The theory's right, but some capos press quite hard on the strings, so when taken back off, strings probably won't be accurate open. It happens the other way too - if a guitar is spot on open, putting on some capos will result in the guitar then being out of tune, maybe only slightly, but still out.
Best bet is either to rely on a tuner which can tell other tunings apart from standard, use a keyboard to match, or learn to tune the guitar to itself, which worked pretty well (for many decades!) until electronic tuners became available. It still works - I use harmonics - but the downside is in noisy places it's not the easiest to hear.
Not for sure. Depending on how much you have to change the tension by turning the peg, and how tight the capo frets, the tuning can change when tension is redistributed by removing the capo.
If you must tune with a capo, remove it and reattached it every time between adjusting or turning a tuning peg and checking the tuning accuracy. That will allow the string a better chance to redistribute tension.