In all the shower installs I overseen, the company I worked for maintained that caulk is to be used in any inside corners except where excessive/standing water is. For example, caulk corners where tub and walls meet, and vertical inside corners where the walls come together. DO NOT caulk where the floor and walls meet, I personally seen caulk creep out of the joint after a week of use in a residence I worked on while I was finishing up the punch list.
Find a reputed mold/mildewcide to clean the joint after you clear all the old grout then allow to dry as long as possible and use a color matched caulk. This is available at Home Depot or at tile supply houses. It is available in sanded or unsanded.
In theory, it may trap moisture, but when the tile is installed and the grout sealed, which should be the case, between the tile being water impervious and the grout has been sealed from moisture infiltration, the amount of water getting past both those points is very minor in my opinion, I read this is so elsewhere.
11-30-2013 addition
To clean the joint of old grout, protect the tub with blue painters tape, maybe a few layers and use a razor knife to begin to V cut the grout out, again be careful the side of the razor knife will be rubbing very hard on the tub/tape and can wear through readily. There is a grout saw available at hardware stores, but they will cut a joint about 1/8" wide, most tile is set closer. If you use this, use a metal shield of some sort to protect the surface of the tub. A large drywall knife (12") would work or a section of very thin gauge metal purchased from a hardware store where they sell small pieces of flat metal. Careful with this, it must be kept secure from moving or the metal will deposit gray marks on the acrylic. Tape it to the tub at the ends and let the metal go up to the face of the tile so any scratches is covered by the new caulk.
To caulk, first the joint MUST be clean and dry, cut the tip to the size of the joint, and apply caulk. Look as the caulk is exiting the tip to ensure the caulk is going INTO the joint, this is where the work is done by the caulk, This why it must be clean. Yes there will be caulk EVERYWHERE to a degree. This is why I keep a 5 gal. bucket handy half filled with water and a rounded edge sponge to clean the surface. Paper towels won't last in the cleaning process, and it may gouge out the caulk if a folded corner of the paper towel goes awry.
To clean the excess caulk, with the wetted sponge, squeezed the daylights out of it to get most of the water out of it. It is imperative you are not depositing too much water on the surface of the tile, it will weaken the caulk. You can clean the caulk from the surface completely leaving the caulk only in the joint and with the color matched caulk, it will look like the other joints on the wall. These caulks are specified to tubs and showers.
Number 1
It could cause a problem, however it it unlikely. Use the shower a few times, and see what kind of condensation settles on the wall above the tiles. If you can see visible droplets of water beading on the wall, you will have a problem over time. As your house settles, and expands/contracts over time the grout will develop fine cracks. These droplets will over time go behind your tiles, and weaken the structure of your wall. If it was cement board it probably wont even matter, and we are talking decades before damage here, but why take the chance for a $5 tube of caulk? There are several brands of caulking meant for the bathroom that can be painted.
Tip - when you apply caulk and run your finger over the caulk to get a smooth surface, make sure your finger is wet first. Makes a huge improvement to how the caulk spreads and looks.
Number 2
The silicone will affect the drying of the grout, and you will probably find it takes well over a week to match the color of other grout. Depending on how good the seal is (i.e. is the grout exposed to air or a porous surface underneath), it could take a very long time. I wouldn't worry about it other than for cosmetic reasons.
Best Answer
There isn't anything wrong with grouting between tub and tile given that you do it right. I fill up tub full of water (weight) and grout. I feel grout is just easier to deal with. My grout has lasted 6-7 years now in one bathroom without any sort of cracking or separation. If there was separation I would caulk it.
It does get mildew and mold and I probably spray it with cleaner once every 4-5 months. That is about it.
From what I can tell from your pictures, that doesn't seem like grout. And if it was grouted it was done very poorly. That looks like a mastic or even a white thinset to me. And then caulked over. It also looks like too big of an area to caulk. I would suggest trying to break out some of that area and then going over it with grout. (Also it looks like crappy caulk. I would use silicone here)