What are the dharmas prescribed for the Grihasthas (family men)?

The Dharmas for the Grihastas are beautifully described in the 14th chapter of the 7th Canto of Srimad Bhagavatam as a conversation between Yudhishtra and Narada. You can refer the same.

If a grihasta is really serious about realizing the Self, and goes through the spiritual process while sustaining the family, then, wouldn’t s/he lose the interest in everything else like family/work etc? How can one continue to lead a normal everyday life, while seeking out to destroy desires?

When we do something, we either do it with attachment or we do it out of duty consciousness.
In office, we carryout the work sincerely because we are bound by duty-consciousness. There is no question of attachment here.
When at home, in family life, we carry out our duties with attachment.
We don’t show as much attachment in our office work as we do in buying a new shirt for Diwali or a new toy for our child at home.
The commitment that is shown in your family life should be limited to that of rendering your familial duties. If you can carry out your affairs at home with the same duty consciousness and no attachment to the activities, then you will be able to progress in spiritual life, while still being in family life.

I am a devotee of Sriman Narayana. I have decided to live a family life and yet be spiritual. Can I see the Lord even though I don’t follow Brahmacharya and am a householder?

Being a householder is not at all a road block to see God face-to-face. History has it that most of the Mahans who have seen God were householders. Continue to chant the Maha Mantra with faith and perseverance. When you incessantly chant the Maha Mantra with patience, you will definitely be able to see Sriman Narayana some day.

How to strike a balance between living in the spiritual world and the “worldly” world till your duties in the “worldly” world are not done?

The life of a householder is the most blessed one.
Even if one is wealthy and leads a comfortable life, if he fails to chant the Mahamantra, he is certain to suffer in the life after death! If a Sanyasi follows the rigorous austerities laid down for him, he suffers in this birth but is happy in the life beyond. By being a householder, leading a righteous life and chanting the Mahamantra, one can enjoy this life as well as in the life beyond!
Is it not a blessing to be a householder?
So chant the Mahamantra. Chant it with faith, love and patience. Then you will see for yourself that the heaven is at your feet, even while doing your worldly duties.