Nayara, the "river" is a stream. Pictures in the Current Conditions thread show it. There is even a way to bypass that first crossing -- I posted links to the description of the old trail just below the second picture. I think in the next few weeks, I'd opt to go up the old trail.

Also understand that "bethjanelle" who made it look so easy is not your average hiker. Note how they caught and passed other hikers -- they were really in shape and fast. Even taking only microspikes is an exception to what most are doing.

Your question regarding gaiters -- yes, you should have them. Call Elevation Climbing in Lone Pine -- see if they rent them. If you "posthole" -- your foot plunges deep into snow because it is soft and slushy, and there is a hollow space underneath -- when you pull out of the hole, your show will scoop a handful of snow inside. After several of these, you will have cold wet feet!