1) How does owning a home fit into my financial portfolio? Most seem to agree that at best it is a hedge against rent or dollar inflation, and at worst it should be viewed as a liability, and has no place alongside other real investments.
Periods of high inflation are generally accompanied with high(er) interest rates. Any home is a liability, as has been pointed out in other answers; it costs money to live in, it costs money to keep in good shape, and it offers you no return unless you sell it for more than you have paid for it in total (in fact, as long as you have an outstanding mortgage, it actually costs you money to own, even when not considering things like property taxes, utilities etc.).
The only way to make a home an investment is to rent it out for more than it costs you in total to own, but then you can't live in it instead.
2) How should one view payments on a home mortgage? How are they similar or different to investing in low-risk low-reward investments?
Like JoeTaxpayer said in a comment, paying off your mortgage should be considered the same as putting money into a certificate of deposit with a term and return equivalent to your mortgage interest cost (adjusting for tax effects). What is important to remember about paying off a mortgage, besides the simple and not so unimportant fact that it lowers your financial risk over time, is that over time it improves your cash flow.
If interest rates don't change (unlikely), then as long as you keep paying the interest vigilantly but don't pay down the principal (assuming that the bank is happy with such an arrangement), your monthly cost remains the same and will do so in perpetuity. You currently have a cash flow that enables you to pay down the principal on the loan, and are putting some fairly significant amount of money towards that end. Now, suppose that you were to lose your job, which means a significant cut in the household income. If this cut means that you can't afford paying down the mortgage at the same rate as before, you can always call the bank and tell them to stop the extra payments until you get your ducks back in the proverbial row. It's also possible, with a long history of paying on time and a loan significantly smaller than what the house would bring in in a sale, that you could renegotiate the loan with an extended term, which depending on the exact terms may lower your monthly cost further. If the size of the loan is largely the same as or perhaps even exceeds the market value of the house, the bank would be a lot more unlikely to cooperate in such a scenario.
It's also a good idea to at the very least aim to be free of debt by the time you retire. Even if one assumes that the pension systems will be the same by then as they are now (some don't, but that's a completely different question), you are likely to see a significant cut in cash flow on retirement day. Any fixed expenses which cannot easily be cut if needed are going to become a lot more of a liability when you are actually at least in part living off your savings rather than contributing to them.
The earlier you get the mortgage paid off, the earlier you will have the freedom to put into other forms of savings the money which is now going not just to principal but to interest as well. What is important to consider is that paying off a mortgage is a very illiquid form of savings; on the other hand, money in stocks, bonds, various mutual funds, and savings accounts, tends to be highly liquid. It is always a good idea to have some savings in easily accessible form, some of it in very low-risk investments such as a simple interest-bearing savings account or government bonds (despite their low rate of return) before you start to aggressively pay down loans, because (particularly when you own a home) you never know when something might come up that ends up costing a fair chunk of money.