"But I think he gives different meditation instructions to different people with regard to the path to reach Nibbana."
Yes, according to current difficulties of his audience. For example:
"Develop the meditation of compassion. For when you are developing the meditation of compassion, cruelty will be abandoned.
-- MN 62
"would like to know what kind of meditation practice did the Buddha advise the monks to do, in contrast to lay people."
I've never seeing this distinction being drawn.
"Did he always teach them to practise Samatha first, and then Vipassana?"
Ven. Ananda said: "Friends, whoever — monk or nun — declares the attainment of arahantship in my presence, they all do it by means of one or another of four paths. Which four?
"There is the case where a monk has developed insight preceded by tranquillity. As he develops insight preceded by tranquillity, the path is born. He follows that path, develops it, pursues it. As he follows the path, developing it & pursuing it — his fetters are abandoned, his obsessions destroyed.
"Then there is the case where a monk has developed tranquillity preceded by insight. [...] Then there is the case where a monk has developed tranquillity in tandem with insight [...] Then there is the case where a monk's mind has its restlessness concerning the Dhamma [Comm: the corruptions of insight] well under control. There comes a time when his mind grows steady inwardly, settles down, and becomes unified & concentrated. [...] Whoever — monk or nun — declares the attainment of arahantship in my presence, they all do it by means of one or another of these four paths.
-- AN 4.170
"Did he ever teach anyone to practice the 'dry insight' path, without ever doing Samatha meditation?"
Hard to tell. But it's reasonable to think that he would not teach samatha to an accomplished jhana meditator, and instead, would focus on what he or she is missing to attain nibbāna.
"As a general rule, were there any differences in the meditation practice that he advised between monks and lay people?"
Not that I know of. I didn't really keep track of who the Buddha was talking to, to be able to answer this conclusively. But there were lay disciples accomplished in meditation during the Buddha's time. For example, Citta, a householder, declares his ability to attain jhanas (see SN 41.8). Though I can't tell if he learned it from the Buddha's teachings, at least it seems unlikely any practice was kept away from lay disciples -- besides, lay disciples were known to be reciters of entire nikayas.
"I am aware that the Buddha prescribed basic morality for all, and that some people would reach Nibbana at the same time as hearing him speak about some aspect of Dhamma without doing any kind of formal meditation practice."
It's generally believed the people who did not made much practice under the Buddha's guidance where already very accomplished.