This is what I am aware of (Hanafi ruling. Thee should not be difference in other rulings)
Makrooh times are Makrooh for salaat (in fact Makrooh for Sajda, and sajda is part of every salaat, hence salaat). During those time, one can do recitation of quran, darood, Zikr, wazeefa, Dua, etc
You can perform Asr Salah, so late that if you get time to even complete on Rakaa't (it might be shorter than that) and Maghrib time kicks in then your Asr Salat is valid.
Once you perform Asr then you cannot perform any other Salat (except Qaza-e-Umaree), till Maghrib Farz prayer
If you missed Fajr, and woke up late, then dont delay it further, even if the sunrise is happening, just go ahead and perform it. If you've performed Fajr on time then wait till 7-10 minutes after sunrise to do Ishraaq, or any other Nafl.
Zawal time is usually +/- 5 minutes (that is just to give it a safe window and eliminate any chances of error). Best way of calulating is learning about the sunrise and sunset time and divide the differene by 2.
Just wanted to add that according to Imam Abu hanifa, if one enters the masjid at the time when Imam is leading the Fajr prayers, then even if you are sure that you would be able to catch him during farz, you shouldnot offer sunnah at that time but join the Jammat and offer sunnah " After the sunrise" NOT just after the farz namaz. ( Though you may offer sunnah at home if you think that jammat would have started but you would be able to catch it). < The reason of this I understand is that one is not able to focus well in praying while the Imam is reciting in a loud voice
Imam Shafi allows to offer sunnah just after the jammat of Fajr namaz.
p.s. If you want to know the starting time and ending time of Fajr Salah and also the timings of other salah in your city kindly check the following site (first do select “English” in the language box on right top. Than select Prayer times on left bottom of the page)
brother.