First, fix the source of the noise!!
Could be as simple as a monitor or dimmer or electrical equipment like a UPS or motor nearby (you'd be surprised the amount and types of noise sources there are around you!)
Even some LED bulbs introduce noise so be open minded in your search
Secondly, sometimes you get ground loops which can be very difficult to trace, but typically if you are getting noise when you have other gear connected, you may have an issue. Way too much on the subject to cover here but you have to at least know if you have one or not.
Thirdly, shield and properly 'ground' your guitar-costs a couple dollars and some time but you will be happier with the results
4th, use less gain! If you're tracking multiple guitars you likely will get better results with less gain than more! Also be mindful of other 'gain' being added - makeup gain via other pedals or drive/fuzz pedals....additional connected equipment = additional noise. The least noisy connections are those in the virtual grid of the axe, but even there you can add a lot of noise (or amplify it rather) by adding drives and boosts. Be careful to not go overboard.
5th, after most of the noise sources are sorted, use a gate or turn down the volume between parts, just like how it was done old school (not ideal)
I use an ISP decimator, but the axe iii is so quiet and has so many gate options that I find I don't need the decimator like I used to. I usually don't use very extreme settings in the axe's gate as they tend to cut off the transient of the note as you mentioned. I'd look at how the threshold is set first...usually a bit lower than I'd like will allow the transient through without being too noisy. Remember that when you start to play that transient noise is not as noticeable as say the end of a decaying note.....so feel free to adjust down the threshold to a little less than you may like and see if that works.
6th, if you have noisy or microphonic pickups you may look at swapping them out completely. Some pickups I've been able to turn the gain up to ridiculous levels and they are dead quiet...not quite the ideal solution as you may already like the pickups you have, but if you are looking for solutions, this is also a viable one.