Did ancient Chinese have tones (声调)?
Linguists divide pre-modern Chinese broadly into two periods: Old Chinese and Middle Chinese. I wanted to preface my answer by noting that Bernhard Karlgren used the term "Ancient Chinese" to refer specifically to Middle Chinese, and it appears that your questions seem to be referring to Middle Chinese as well, though I will be making a note about Old Chinese in my answer.

Middle Chinese is documented as having tones (in particular by the Qieyun in 601) while Old Chinese has been postulated as being atonal. I will go into more detail as I address your questions in order:


The tone contours of the various modern Chinese varieties can appear to be superficially quite different, but the tones themselves are actually quite consistent in the way they evolved from the tones of Middle Chinese. Words that belong to particular tone categories tend to remain in the same tone category across the various modern Chinese varieties, even though the ways the tone itself is pronounced has diverged among differing dialects.

Take for example, the word 平, which generally always has the 陽平 tone across all of the modern Chinese varieties; however, the 陽平 tone itself has evolved into having, for instance, the contour 35 in Standard Mandarin but contour 21 in modern Cantonese.

Various tone categories can end up getting split or merged though during the evolution of any particular Chinese variety due to various phonological environments, so for instance, Middle Chinese 陰去 and 陽去 tones both ended up merging into one 去 tone in Mandarin (with contour 51), while they remained separate in Cantonese (with contours 33 and 22, respectively).

One area where tone mergers were not very consistent is the loss of the checked tones (入) in Mandarin. Words with this tone ended up being redistributed to the other 4 modern Mandarin tones seemingly randomly in Beijing Mandarin (and thus also in Standard Mandarin which is based on Beijing Mandarin); however the redistribution in other dialects of Mandarin appear to more consistent in this regard (refer to the table in this article). One explanation for this inconsistency is that checked tone words in Beijing Mandarin ended up being pronounced in a variety of different tones once the distinguishing feature of checked tone words (namely the stop consonant codas) were lost and the confluence of various dialects in the capital reshaped the tones with which these words were pronounced in different ways.

Refer to the table in this article for how the various modern Chinese varieties map the original tones of Middle Chinese into their modern tones and tone contours.
Japanese and Korean belong to separate language families that are not related to Chinese even though they imported many Chinese words. The time period during which most Chinese words were imported fall squarely during the Middle Chinese period, which was documented as already featuring tones. Japanese and Korean though, are not tonal languages, and because they don't distinguish lexical tone, it's not unexpected that they would also not distinguish tones in the words they imported.

For a parallel example, English is an atonal language that has also imported a few words of Chinese origin, such as bok choy from Cantonese and feng shui from Mandarin, and the tones with which these words are originally pronounced are similarly ignored in English.
As I mentioned above, the words with stop codas were classified under the checked tone in Middle Chinese, and when the stop codas were lost in Mandarin, their tone contours ended up being redistributed to the other tones. Strictly speaking, yes, this did end up resulting in a loss of information leading to a possible increase in ambiguity; however, this is not an uncommon occurrence in the evolution of any language. Distinguishing features have been lost in the history languages across the world, but their speakers generally make up for it in other ways. Mandarin, in particular, lost many of the distinguishing features that were present in Middle Chinese, leading to a reduced syllable inventory, but it has compensated by increasing the number of disyllabic words in its vocabulary.

It is interesting to note, though, that loss of certain codas in the transition from Old Chinese to Middle Chinese did indeed get transformed to tones. It is postulated that Old Chinese was actually not a tonal language, and words ending in a glottal stop (-ʔ) in Old Chinese ended up cheshirizing to the 上 tone and words ending in -s to the 去 tone. See this article for more details.

Similarly, a later tonal distinction caused by voiced vs. unvoiced initial consonants in Middle Chinese lead to the split of the four 平上去入 tones into two registers, 陰 and 陽, leading to 8 tones (this marked the transition from Early Middle Chinese to Late Middle Chinese). The voiced vs. unvoiced distinction has disappeared in most modern varieties of Chinese (a notable exception being the Wu dialects), but the tonal distinction still remains in many of them.