I try to answer it. there are slight changes as how should one pray depending on which Imam one follows. All imams are right. A Hanafi will go as following:

Namazi jis bhi rukan main Jamat se mile ga us par Imam ki atat lazim ho gi.

Suppose, its Isha prayer, 4 faraiz rakat, and i am late and join

a) 1st rakat before ruku or during ruku so that i am able to say tasbeeh three times, then the rakat is offered, no need to repeat. Imam kay sath salam phairunga.

b) If I join 1st rakat after ruku or during ruku so that i am unable to say tasbeeh for three times before Imam says Samiallaholimanhamidah, then i missed 1st rakat and have to repeat after imam's conclusion (salam)

How will I complete 1st left over rakat? I will repeat it by stopping tashahud in the fourth rakat before darud sharif and wait for imam to say salam. When he declares salam I will stand up and say that rakat as normally mufrid namaz, means SuraAl Fatiha + other sura and will complete it normally.

c) if i miss 1st rakat altogether and in the second rakat before ruku or during ruku so that i am able to say tasbeeh three times, then the second rakat is counted, only i have to repeat 1st as above.

D) if i join 2nd rakat after ruku or during ruku so that i am unable to say tasbeeh for three times before Imam says Samiallaholimanhamidah, then i missed (1st + 2nd) and have to repeat in the end.

How to repeat 1st and second: i will simply stop after Tashahud in the last rakat of imam i.e. 4rth one and would not offer darud,dua but will rise (qayyam) and continue 1st rakat (then again stand w/o qa'da), offer 2nd rakat (in the end of which i will say salam and conclude namaz)

e) agar main 3rd rakat main ruku se pehle ya ruku k darmiyan jamat se milta hun aise k main teen bar ruku tasbeeh parh laita hun to teesri rakat count ho gai, sirf 1st + 2nd missing hain jis ka tareeka uper bataya.

f) agar main 3rd rakat main milta hun to agar tasbeeh mukammal nahi ki ya ruku ke bad milta hun meri 3rd rakat ada nahi huwi.

Now I will say 1, 2, 3 rakat after imam says salam. What i will do is, i will stop after Attahiyyat in the final rakat and would not offer darud and dua.but will rise in qayyam to resume left over rakaat. now since qa'da/Tashahud has to come alternatively and it has to come in the final rakat i.e. 3rd, so it will also come in the 1st. So when i stood to offer 1st left over rakat, i recited S.Fatiha+any sura, will have qa'da, but after
Attahiyyat i will stand again for 2nd rakat (same S.Fatiha+any other part of Holy Quran) which will go w/o Attahiyyat, and then 3rd (only Fatiha in it since farz namaz main 3rd aur 4th rakat main surat nahi milate) which is the final rakat and will conclude namaz with salam.

Summary: 1st (S.Fatiha+other sura, with attahiyat/tashahud only) 2nd (S.Fatiha+other sura, no qa'da/tashahud) 3rd (final: S.Fatiha only and uptil darud dua and salam)

g) Agar main 4rth rakat main jamat kay saath milta hun to agar ruku main ya is se pehle yun mil gaya keh tasbeeh parh li imam ki injunction se pehle to 4rth rakat has been counted. Only then 3 missed, jisk tareeka uper likha hai.

h) agar 4rth rakat bhi miss ho gai to mujhay charo paRhani hongi. albatta agar main imam k salam kehne se pehle shamil huwa to muje jamat ka sawab mil gaya (which is 25xinfiradi namaz of Dhuhar, Asr, Maghrib time; & 27xinfiradi namaz of Fajr and Isha)

1,2,3,4 left over hain parhni hain. Inhain usi tarah mukammal karen gay jaisay infiradi farz namaz. 2nd aur 4rth main qa'da karengay. 2nd main darud se pehle uth jaingay, 4rth main darud, aur dua k baad salam pher daingay.

1 (S.Fatiha+other sura; normal) 2 (S.Fatiha+other sura; qa'da) 3 (S.Fatiha only; normal) 4rth (final: S.Fatiha only with tashahud, durud, dua and salam)

Maghrib ki namaz ka tareeka mukhtalif hota hai, sirf thora sa. Agar maghrib ki 1st mis ho gai, to wo normal tareekay se ada karengay. Agar do miss ho gayin, to 1st main bhi qa'da (do zano baithna) aur second main bhi qada, albata 1st chonkay final rakat nahi hai, darud se pehle uth jaya jayega. this is important. yani yahan per alternate qa'do ka rule apply nahi hota
agar 3 ki 3 miss ho gai to mufrid namaz ki tarah mukammal karengay. 1st (w/o qa'da) 2nd (with qa'da but w/o darud-o-dua) 3rd (final rakat: with qa'da, with darud-o-dua and salam)

PS: jamat main shamil hone k liye haste kabhi nahi karni chahiye chahe rakat miss hi q na ho rahi ho, jaise aksar log masjid main bhagte bhi hain, is se masjid k taqadus disturb hota hai. mazeed zaruri nahi k niyyat lafz ba lafz ki jae, agar is se der hoti ho to dil main bhi ki ja sakti hai.

do tarah ki namaz hai, sirri aur jahri. Dhuhar aur asr imam tilawat nahi karta, u know much in islam depends on time of the day. its day's white when these prayers are offered. but why exactly then i dont know because its tradition of the Prophet (SAW), a tradition which was never broken even for one time in these 1433 years. So its true and we have to follow.

Rest assured, aisa sawal competitive exam main nahi ae ga. Seekhne walo ki ghalatiya muaf hain, and Allah knows better.