Is there a recommended intention one should make when giving sadaqah?
It is clear, the intentions of giving Sadaqah can be many and the reward will be increased with many good intentions (Niyah). Some of the intentions can be --
I am giving him these, seeking pleasure of Allah, because I know, Allah likes the deed of giving sadaqah to poor and needy people.
He is a creation of Allah and I am giving charity to one of His creation seeking His pleasure. Because, Rasul (Salla-Allahu-Alaihi-Wa-Sallam) says, "be kind to the creation in this world, the creator on the heaven will be kind to you" (translation of the meaning)
Your heart become soft by seeing the hardness of their life and thus you are helping them. It is one of the good intentions.
If you are giving Sadaqah to a relative of your's, your intention can include "The prophet orders to maintain relative bondings and strengthen it, And I am obeying one of the command of our Prophet"
The Prophet (Salla-Allahu-Alahi-Wa-Sallam) ordered to be good to parents and says one of the way to be good to your parent after their death is - maintaining good relation with the friends and relatives of your parents. You can include this intension on such condition.
There are many more good intentions and you are recommended to include as many as possible to include them in your intension. The reward will be multi-fold with those intentions. Allah said in Quran:
The example of those who spend their wealth in the way of Allah is like a seed [of grain] which grows seven spikes; in each spike is a hundred grains. And Allah multiplies [His reward] for whom He wills. And Allah is all-Encompassing and Knowing source
I would like to hint to the bad intentions as well, just as a reminder.
A bad intention can include Riya (Doing good with the intension of earning worldly benefit). Allah gives great warning to these kind of people in the Holy Quran.
O you who have believed, do not invalidate your charities with reminders or injury as does one who spends his wealth [only] to be seen by the people and does not believe in Allah and the Last Day. His example is like that of a [large] smooth stone upon which is dust and is hit by a downpour that leaves it bare. They are unable [to keep] anything of what they have earned. And Allah does not guide the disbelieving people. source
You should not give sadaqah, so that you can win over him at a later time or injure him with your words. This is indeed very bad. Allah says, This kind of Sadaqah has no reward and it is not considered a good deed.
Kind words and forgiving of faults are better than Sadaqah (charity) followed by injury. And Allah is Rich (Free of all wants) and He is Most-Forbearing. source
Intension of earning any worldy benefit from the Sadaqah.
Your second was this -
Is it standard islamic practice (or is this cultural) to give sadaqah on behalf of another person to add barakah and/or make their life easier and/or remove some calamity from their life? i.e. Can I give sadaqah on behalf of a troubled family member? In the hope that this troubled family member will receive the blessings of me giving the sadaqah on his/her behalf?
Yes, of course. You can give sadaqah on behalf of a troubled family member, you are encouraged to do so as a means of Tightening relationship, which is given high importance on Islam. The prophet was asked a similar question:
Aishah (May Allah be pleased with her) reported:
A man said to the Prophet (sallallahu 'alaihi wa sallam): "My mother has died suddenly. I think that if she were able to talk (alive) she would have given in Sadaqah (charity). So, if I give Sadaqah now on her behalf, will she get the reward?" The Messenger of Allah (sallallahu 'alaihi wa sallam) said, "Yes (she will be rewarded for that)" source