It's probably too late now, but I always take the bottom grill off before I try removing the filter. With the grill on I can't get enough leverage to turn it - but now that you've broken the handle it's probably too late. You could try calling your fridge manufacturer and see if they have any suggestions.

If your softener installer actually placed a layer of GAC (granular activated carbon) in the SALT tank, then he has absolutely no clue whatsoever about how softeners work. The only water that might filter a little bit would be the water used for the brine draw portion of the first few regenerations after installation. If, however, he placed a layer of GAC in the resin tank (which is more likely), then he did you an even worse disservice. Placing a layer of GAC in your softener is a terrible idea for a variety of reasons. GAC (granular activated carbon) and softener resin have very different densities, requiring different backwash rates. One or the other (or potentially both) won't be adequately backwash. Loose GAC should be backwashed daily to prevent compaction and damage. The damaged particles cause increased pressure drop, followed by channeling, resulting in poor performance for the softener. GAC is sharp and damages softener resin, which harms softener performance. In order to work effectively, a GAC bed needs to be WAY deeper than is possible if it is placed in a layer in your softener. This means it won't do much. GAC effectiveness reduces over time, at a rate much faster than softener resin, meaning it will need to be replaced. A thin layer like that - you'll get a year max. Unfortunately, since its mixed up with your resin, that means a total replacement of the softener resin as well. GAC takes up space in the softener tank that is supposed to be empty so that the resin bed may be adequately expanded during backwash. If that expansion is not possible, broken beads and other particulates build up over time, negatively impacting softener performance.

We have a "whole house" filter on the cold water line that runs to the kitchen (from our utility room, which is where the filter is located). It filters the cold water at the faucet, the water line to the fridge, and the pot filler faucet. It also filters the water going to the newly installed faucet outside, right over the deck under the kitchen window, but this was not by design--that faucet taps into the cold water line under the kitchen sink. Theoretically we could use that filter for all the cold water (that's what it's meant for), but we don't. The filter is by CUNO--AquaPure (the filter housing we have utilizes the APS117 and AP117 filter cartridges--the "S" cartridge removes lime scale and we put that one in when we replaced the original cartridge last month). If you go to the CUNO and AP distributor websites, you can find various filter options. I know when I called CUNO to ask about the two diff cartridges the woman in cust/tech support seemed surprised that we had a whole house filter, but that's what our plumber installs. I'll bet they can help you find the right filter for your needs. FWIW, we don't have a separate tap/faucet for drinking water. We tend to get our drinking water from the fridge dispenser which is double filtered anyway (once by the house filter and then again by the one in the fridge grille).

Most places have great water. Some places don't. And that's hopefully something you know about if you live there. It could be organic contaminants (e.g. pesticides, drugs, solvents, ...), it could be minerals (e.g. hard water, "dirty" iron deposits, ...), it could be other industrial inorganic contaminants (e.g. heavy metals, nitrates, ...), it could be biological contaminants (e.g. viruses, bacteria, ...), it could be radon gas, or it could be harmless nuisance impurities (e.g. chlorine smell, sand, ...). Some of these are more likely in municipal water supplies, others are more common in local wells. If you know what you are concerned about, then you can look for the appropriate type of filter. Quite frequently, a big activated-charcoal cartridge takes care of things. And that's the most commonly sold filter you'd get. But for other types of contaminants, charcoal is going to do zilch. Also, it really depends on whether this is just a nuisance contaminant or whether there are real health risks associated with it. If in doubt, have the water tested by a lab. The supply in your house might very well have different water than the one your neighbor has, because internal piping materials differ.