According to Wikipedia and this notes from HKU, the indirect utility function is part of consumer theory, and is defined as the maximum utility that can be attained given a consumers' money income, and goods prices. It reflects both the consumer's preferences and market conditions. It is a model which helps to understand how consumers optimize their expenditure considering the price of products.
The model considers a consumer who faces a world with only two products with fixed prices and one limited budget, and is represented by a perpendicular axis graph, with each axis representing the quantities of each good, and both axis converging on quantity 0. A descending diagonal line is then drawn to represent all the possible quantities of good A and good B the consumer could buy by spending his entire budget. This line is called the goods bundle line, the price of both goods is reflected in the steepness of this line.
Another descending line is then drawn called the marginal utility function, this line tends to be concave and not straight, due to the theorem of diminishing marginal benefits. The line represents all the combinations of quantity of good A and good B that give the consumer the same level of benefit. In fact you could draw a whole map of concave lines, all parallel to each other, and each would represent the total quantity of good A and B needed to achieve a certain fixed level of "satisfaction" by the consumer. This map of lines would represent the consumers' absolute "tastes and needs" sort of speak.
The consumer, in consumer theory, will try to select a bundle which combines just the right amount of both products, and will spend the whole entire budget on doing so, in order to obtain the greatest possible benefit in his consumption process. The decision of how much to spend on each product is represented in the graph by the intersection of the goods bundle line and the one marginal utility function it intersects tangentially with. Any other products bundle in the goods bundle line will not be as beneficial to the consumer as the one where the goods bundle line tangentially intersects one of the utility functions.
This is how you can finally build the indirect utility function, by fixing the price for both goods and making the budget a variable. Now for each level of budget you´d get one single point in the goods bundle line where the benefit is maximized for the consumer, and if you put together all the dots, one representing the optimal goods bundle for each level of budget, you'd construct the indirect utility function. It is called "indirect" because of its' indirect correlation with the price of goods. Budget level thus now becomes irrelevant, and what molds the consumers decision making are prices of goods in the market and the consumers' preferences.
So my answer to your first question, "In general terms, how does one go on to select a given functional form.?", would be that I don´t think the techniques for figuring out a particular utility function, for one or a group of specific individuals, belongs to the economics realm. Because it would involve practical modeling and not just the economic theories any more. I'd say that question could find a better answer in the finance realm.
My answer to your second question, "Are there a class of utility functions for
households that are better than others?", is yes. Consumer theory describes one utility function for any given person and price level of goods A and B. But if the price of any of the products in the consumer's bundle changes, the utility function reshapes given the consumers preferences. Also if the consumers' preferences or needs towards any of this products change, that would result in the reshaping of the utility function.
My answer to your third question, "How does one even judge better, in this context?", is you can't judge better in reality. Perception of the benefit for consuming is subjective, and this subjectivity is embedded in a consumers' map of marginal benefit functions, and through it in the resulting utility function. This makes it impossible to quantify and thus to measure. Making it only possible to estimate. So people use this theory to understand the phenomena in a general level, like predicting the behavior of entire economies, or a whole segment within them.
And for your last question, "Does the selection of a utility function depend on the context and analysis at hand?", my answer is yes; and I quote from this book: -What exactly is a “good”? The answer lies in the eye of the modeler. Depending on the problem to be analyzed, goods might be very specific, like tickets to different world series games, or very aggregated like food and shelter, or consumption and leisure. -
In any case I strongly suggest you specify if your question is theoretical or practical, in order to get a more accurate answer.