At the tiled wall surrounding the tub, at the bottom of the tiles where the tiles end and intersect the tub rim, should it be grouted ( epoxy grout)? or leave it open and caulk with silicone caulking? Please advise!
Matching sanded or unsanded caulking is a must. 9/10 times you can purchase a caulk that matches the grout. By using the matching caulk, it creates a natural looking flow for all the tile and grout you just installed. I recommend what one of the other members suggested and fill the tub half way with water, then use matching caulk around tub, then drain. You’ll have a nice looking tiled shower/tub after
Most bathrooms I've been in seem to have silicone sealant in this area. I think this may be an evolution of tile grout (and other sealant types) which (used to?) be somewhat prone to developing black mould (or cracking, leaking etc), especially where there's a shower over the bath. Some silicone specifically claims to have anti-mould features, although water-fastness is probably its main defense (eg. https://www.amazon.co.uk/2079356-Anti-Mould-Silicone-Bathroom-Cartridge/dp/B01G3OCFVI).
In my own bathroom, the tiler fitted the tiles, with grout to the top of the bath tub. Then, they siliconed around the bath tub (with plenty of water in it at the time). They didn't use a particularly large bead of silicone though, and in some places it's insufficiently covered the gaps, and so there is a tiny bit of water ingress where silicone meets tiles. Over time, this goes black, which only seems 'solvable' by bleaching.
I believe the solution here would be to scalpel-cut out the existing silicone and replace with new (and use a slightly larger smoothing tool), whilst making sure that the silicone gets properly "pressed" into all the gaps between tub and tiles so that it adheres to all surfaces properly.
IMHO, silicone is both great and terrible. Using a smoothing tool (eg. https://www.diy.com/departments/diall-smoothing-sealant-tool/1642133_BQ.prd) is a must, as is wearing gloves and having a selection of rags and whatnot available for wiping up excess (which needs white spirit). You also don't seem to be able to buy silicone in small quantities (and you'll need a sealant gun too). For the professionals, I'm sure none of this is a concern, but for DIYers, it takes a bit of practice to get it right (and probably a fair bit of waste afterwards).
I was always taught that you should use a flexible sealant at every change of direction or material.
So all intersections, angles etc. plus where tile meets another material such as the tub or a non tiled part of the wall etc.
Obviously sometimes this means you'll need a paintable sealant (tile to wood or plastered wall/ceiling for example).
I always use a high quality sanitary silicone between the tile and tub.