People often pray to God vowing to perform something in return when their prayers are fulfilled – like visiting a certain temple/deity, offering money to a temple, circumambulation etc. Sometimes they are not able to fulfill their vows due to illness, accident or other difficulties. Can something else be offered as an alternate, in such cases?

When vows are taken, they have to be fulfilled exactly as they were taken. If a person has taken a vow and is incapacitated that he cannot fulfill it by himself, some one else should take permission from him and fulfill it on his behalf. If he dies before his vows are fulfilled, he will take another birth to fulfill the vows, even if someone else fulfills the vows on his behalf after his demise

I avowed to do some regular offerings to Tirupati Perumal and have been doing it for quite some time now. However, looking at the so many poor and needy in the country, I am thinking of using this money for charity. I hope the Lord will not object to it.

All vows should be fulfilled exactly the way it has been promised. Here, I am reminded of an incident relating to Maha Periyava (Sri Chandrasekarendra Saraswati Swamiji of Kanchi).
Once a lady came to have the darshan of the Mahaswamigal told Him that He appeared in her dream and asked to give Him a golden bowl filled with dry grapes. She then asked His Holiness, if this prayer could be fulfilled by offering it in the form of money equivalent in value to His Holiness.
Maha Periyava refused and asked her to do it in the same manner as has been instructed to her in her dream. Being an affluent person, the lady obeyed by bringing a big bowl made of gold filled up with dry grapes and offered it to the Saint.
His Holiness accepted the offering and took the dry grapes. Turning in the bowl to the Mutt Manager, He asked the bowl to be handed over to the government immediately, the reason being that, that was a time when the ‘Gold Control Act’ was in place according to which, it was an offence to possess articles in gold other than ornaments and that too was restricted by an upper bound laid by the law.
Hence the prayers have to be fulfilled in the same manner as promised.

I am not able to perform regular prayers in the morning (due to my work schedules) nor in the evening (as I am so much tired after I return from work). This gives me a feeling of guilt of neglecting prayers. However I listen to bhajans when I drive that gives me a great peace of mind and satisfaction. My question, now, is, Are proper prayers a must?

Although certain times are considered best for prayers, the key logic behind giving certain time restrictions and rules and regulations to performing prayers is only to enforce people to perform prayers. Otherwise, in general, there are no time restrictions or compulsions for personal prayers. You can do it anywhere, and at any time. Truly, Bhajans are prayers only.
Conversing with God in a personal manner is prayer in itself. Thanking God for all the good things He has given you is prayer in itself. Singing His Divine Name is prayer in itself. Mere hearing of Bhajans is a prayer in itself. So your guilt is unfounded.

Whenever I visit the temple or pray at home, I just chant some Slokas I know. I forget to mention the trouble or thank for the joy the Lord gives me. Sometimes I wonder if it is a right way to pray. Please advice.

It is really heartening to note that you chant Slokas forgetting everything else. This is the way to chant and we all aspire for that state. So please do not doubt the rightness of this way. While you chant the Slokas with one-pointed mind, your Slokas will do the necessary prayers for you.

I don’t pray to God for material benefits. However I pray to God and chant during times of distress. Is this true Bhakti? Pray tell me what true Bhakti is.

There is a rare type of herb called ‘Sparsha Vedi’ The greatness of this is that, when a piece of iron is touched by a leaf of this herb, it turns into gold. Likewise, when a person, however impure s/he may be, is touched by the initiation of a Guru, he is purified.
So do not worry. Stop analyzing yourself. Believe only in the grace of God, have faith and take life as it comes. Keep chanting regularly. That will verily protect you.

My close relatives tell their problems to me. I pass it on silently to my Guru and God and sincerely praying to them to solve their problems. Am I right in doing so?

You are not only doing the right thing, you are doing the best thing!
Make them understand that you are praying to your Guru and God for your welfare and that it will definitely perform miracles in their lives. Ask them not to worry and request them to do the same thing. Talk to them in a manner that will console them.