After Azan, how long should I wait before I perform Salah? Is it Haram if I delay prayer for no very important reason? Another question I have is: is it Halal for me to have a photograph of someone special to me by my bedside.

Praise be to Allah, the Lord of the World; and may His blessings and peace be upon our Prophet Muhammad and upon all his Family and Companions.

1) If you can not perform a prayer with congregation in a mosque due to some genuine reason of Sharia (a legal reason), then you are allowed to delay performing the prayer for a period of time that is needed to answer the call of nature, to make Wudu and to perform the prior Sunnah of the obligatory prayer. This, in fact, is the time of superiority or the optimal time. Moreover, performing the prayer between the above-mentioned time and the end of its fixed time is licit except concerning the prayers of al-'Asr and al-'Isha. Since delaying al-'Asr till sun becomes yellow-and al-'Isha till after midnight is detestable. Performing these two prayers after the above-mentioned time is called the time of permissibility, while performing them before that is called the time of preference.
Anyway, performing these two prayers in both times mentioned above is sufficient. But, performing them instantly at the beginning of their fixed time is the optimal choice, since the Prophet (Sallallahu Alaihi wa Sallam) was used to performing prayers at the beginning of this appointed times unless there was a genuine reason.
In fact, there is a danger for one who is used to delaying the performance of prayers that his/her faith becomes weak and belief decreases due to being accustomed to delay and following only what is sufficient.
As for delaying the prayer until its appointed time has finished, without a genuine reason, is Haram and a doer becomes sinful and should be punished unless he/she repents and returns to Allah, The Great and Almighty.
Allah Says (interpretation of meaning): {Then, there has succeeded them a posterity who have given up As-Salât (the prayers) [i.e. made their Salât (prayers) to be lost, either by not offering them or by not offering them perfectly or by not offering them in their proper fixed times, etc.] and have followed lusts. So they will be thrown in Hell. Except those who repent and believe (in the Oneness of Allâh and His Messenger Muhammad SAW), and work righteousness. Such will enter Paradise and they will not be wronged in aught.}[19:59-60].
The commentators of the Qur'an said: neglected prayer means they do not perform prayers in their fixed times.
2) If a picture you mentioned is embossed, i.e. art-dimensional one or a portrait, then it is Haram to make it or to use it. It is confirmed that makers of such portraits and those who use them in their houses will face a strong threat of punishment from Allah.
But, we hope to be right, if such pictures are only photographs, there is no objection to use them according to the predominant opinion among Muslim scholars. They do not constitute a form that is similar to Allah's Creation. Here are two conditions that should be met: (1) such a picture should not be with any form of glorification; and (2) it should not be a forbidden one such as a picture which demonstrates some of the private parts of its owner. Anyway, it is better to be far above such a picture, since it is not one of man's needs or necessities. Our last opinion is to help you be outside the difference among scholars.
Allah knows best.