As Andy said, your best approach will be to read the label.

I don't think there's a "one size fits all" answer to your question. The reason is that different products act differently.

Granular fertilizers will often say to apply when the lawn is dry and to water in after application. Granular weed killers often work best if the grass is wet first and not watered for some time afterward. The weed killers in weed and feed products usually make their application more like weed killers than fertilizers.

As Andy said, liquids usually work best on dry lawns. After using them, some need to be kept dry, while others need to be watered in.

As you can see, there are too many variables for us to answer in generalities. If you can get specific products, we can look into those and let you know (but we'd probably just be finding the labels online and reading them).

A general question may enable everyone to get general opinions from experts. Since you asked specific products, I am going to narrow to two popular granular and liquid of herbicides you mention the most, LESCO Turf Fertilizer 24-0-11 and Weed B Gon Max Plus Crabgrass Control. Assuming the granular of LESCO is spread first and then the liquid of Weed B Gon is sprayed, how shall we apply them, before or after rain or no rain at all?

If you take the hint from how a houseplant is given fertilizer---a liquid type of course--we water the plant first, then the fertilizer. The water draws the fertilizer down to the roots.
Usually not as much liquid is involved when fertilizing and so we water to make sure the food gets down to the roots.

Same applies to the lawn. Water it first, the liquid percolates down to the roots and when the weed killer is applied it follows the liquid. Usually about a 2-hour wait is recommended between applications.

Dry granular fertilizers are usually recommended to be given water AFTER application. It drives the granules off the blades and down into the soil where its action is wanted.

Weed and feeds are no different. Water afterwards.
Liquids, water before.

And don't apply a water soluble when rain is forecast in the next 24 hours.
And don't apply in a hot sun when it can defeat the whole purpose.

"A general question may enable everyone to get general opinions from experts."

But there isn't one answer for all cases here.

I couldn't find a label for the Weed B Gone product, but I think it's a liquid. You want to get the weeds growing well before using it and spray it when the leaves are dry.

From the label for the Lesco fertilizer:
"The best results with this product are obtained
when it is applied to dry turf and actively growing grass. Water into the turf soon after application. Avoid mowing immediately following application to prevent pick-up."

Since Jeannie and I gave different advice about the weed and feed, I thought I'd double check.

Here is the label.

It doesn't allow copying and pasting, but it says to water thoroughly several days before applying to sustain moisture until the next watering. It also says to apply it when the leaves are moist from dew, rainfall or irrigation and that if the leaves are dry, you should sprinkle lightly before using it.

Three different products. Three different answers for when and how to apply and time for rain and watering.

It seems that the opinions are different on applying the liquid of Weed killer. Let me ask one more time. Should the liquid of weed killer be applied before or after the rain? I also wonder whether the urea is better than the bag of fertilizer such as LESCO Turf Fertilizer 24-0-11 for the lawn care program and why?

I didn't see any differences on the liquid. Apply it when the weeds are growing well and the leaves are dry. And try to keep them dry for a few days. If you want to increase the effectiveness, add a couple of drops of dishsoap.

Whether the urea or the 24-0-11 is better depends on whether you need K. If you need K, the 24-0-11 is better. If not, the urea is better. One potential problem with urea is that it's easy to apply too much and cause nitrogen burning. You use roughly half as much urea as 24-0-11.

By the way, is it possible to roughly know whether the K is needed or not without the soil test?

"By the way, is it possible to roughly know whether the K is needed or not without the soil test?"

Not usually. I live in an area where all the soil is high in K, so I don't need a soil test to tell me that. But I'm the exception. Unless you live in an area where the soil is known to have high K, there's no way to know if you need it. If you live in an area with low K, you may not need it because a previous owner may have applied it.