Improving Prayers and Offering Missed Ones

I do not really understand what you mean by "Qaza-e-Omery" prayers. However, if you mean in general a prayer after you have missed it, then I would say that the prayer is the same as you would have offered had you not missed it. So if you missed Zuhr1 prayers, then when you offer it later, you would offer four obligatory rakats. That would fulfill the required part. You can obviously offer sunnah and nafl prayers if you like.

With respect to Qaza2 prayers, I would like to add a word of caution though. It seems that there really is no concept of 'Qaza' prayers as such in Islam. A prayer must be performed on time in normal circumstances barring two conditions only: 1) prayer is missed because of forgetfulness, 2) one sleeps through the prayer. In both these conditions, which should be rare for a good believer, the prayer is to be performed as soon as one realizes that they have been missed. Of course, extraordinary circumstances may have to be dealt with differently.

Having said that, I believe that Islam does lay immense importance on offering prayers on time and every good believing Muslim must do so in order to grow them spiritually and get closer to God Almighty. It seems that you are already involved in prayers and trying your best to get close to the Almighty, so the more the effort you put in, the better it will be.

As for the second part, when you are offering prayers, try your best to feel as if you are standing in front of God Almighty, asking Him for His help and that you are being answered when you offer your prayers. Doing physical actions is not the only requirement, prayer should also help you get spiritually closer to God. If both these conditions are met, there is no reason to believe that your prayers will not be accepted.

In addition to that, to further improve the experience of the prayer, you may also find it very useful to learn the meaning of Al-Fatihah (the first chapter of Quran that we recite in prayers) and bring the meaning to your mind when you actually pray. It will make the prayer much more involved for you. If you can learn some chapters from Quran with meaning so you can recite those in the prayers too, that would also be very helpful.