Having the animations and the visuals together at the last minute is part of the game development process. Unlike film, the idea of post production phase in the traditional sense doesn't really exist. Working without or with limited visuals is an important part of the skill set of a game audio designer, what you are effectively having to do is work in parellel with all the other disciplines.
There are 2 main challenges for getting the audio correct. Firstly like you mentioned is conceptualising the sound, the second is getting the sounds in sync. For conceptualising what we do at Remedy is firstly gather all the information on the feature we can, concept art, scripts, game design documents, then we sit down with the creative director and art director and come up with a collection of descriptive words, we usually come up with 2 or to describe a feature, e.g. cold, shocking, quick, brutal etc. This at least ensures we are all using the same vocabulary. Then we compare that description to how it fits with other descriptions of other features. From this you kinda get a balance of how the sound you are designing fits into the game. Then we go and create a collection of sounds that fit the description, and compare them against the visuals as they are developed. Sometimes it can turn out that the audio drives the visual design. The more you work with a team the better you can guess what really going to happen ;).
Sync is another issue - another thing I try to determine very early on is how changeable a feature is. e.g. is it something that will be baked and then left alone, or has it got lots of tweakable variables that will change into the last second. If it is baked then we can design sounds that sync offline working more like syncing to picture, syncing part is done really late in the process. The variable we usually granulate the sound design and design systems that stitch it all together at runtime. This means that whatever the designers change the sound design will follow it automatically. With this system we try to get our sounds in as early as possible.
Hopefully some of this helps.