runtime of problems vs algorithms
Yes it can. When you take RAM model and your algorithm has a lot of multiplications (dominant operation), your multiplications take unit time. Even if the numbers are $2048$ bit numbers. But if you take multitape Turing Machine as your model, the cost of multiplication matters and will be included in complexity.

Algorithm is high level description, implementation will vary. Getting back to multiplication, algorithm describes that in one step you have to multiply two numbers, but if numbers are bigger than computer words instead of writting "*" you have to implement it.