Axler's approach puts the cart behind the horse where it belongs. Learning to do row reduction by hand is a really pointless exercise that misses the forest for the trees. But if want want the nitty gritty on computation, Golub and Van Loans "Matrix Computations" is pretty classic.
Only if you want to be a real mathematician. Most people don't and are better served by learning to calculate, since that teaches you how the techniques of linear algebra are applied.
You actually do need to learn to do row reduction by hand because it teaches you what is going on. Same reason you need to learn to integrate by hand even though you'll never do it again after your coursework.
BTW, I learned linear algebra from Axler first, so I have some basis for comparison. There is a reason Axler is not considered an introductory text even though it is not very hard.