Use nested #replaces.
Is it possible to replace multiple different strings within one string?
For example, I would want to do {{#replace:The dog is jumping.|dog,jumping|cat,walking}} or something similar to receive the output ”The cat is walking.”
Is this possible in any way?
Yes, just nest #replaces: {{#replace:{{#replace:The dog is jumping.|dog|cat}}|jumping|walking}}
Just be aware of the expansion depth limit (to say nothing of code readability); if you need a lot of separate replaces on the same string, it will probably be better to write it in Lua, as a Scribunto module. (You could also use Extension:Variables, but that extension unfortunately has an uncertain future given the direction the parser is headed in.)