Note: I am making a USA-assumption here; keep in mind this answer doesn't necessarily apply to all countries (or even states in the USA). You asked two questions:
I'm looking to buy a property. I do not want to take a risk on this
property. Its sole purpose is to provide me with a place to live. How
would I go about hedging against increasing interest rates, to counter
the increasing mortgage costs?
To counter increasing interest rates, obtaining a fixed interest rate on a mortgage is the answer, if that's available. As far as costs for a mortgage, that depends, as mortgages are tied to the value of the property/home. If you want a place to live, a piece of property, and want to hedge against possible rising interest rates, a fixed mortgage would work for these goals.
Ideally I'd like to not lose money on my property, seeing as I will be
borrowing 95% of the property's value. So, I'd like to hedge against
interest rates and falling property prices in order to have a risk
neutral position on my property.
Now we have a different issue. For instance, if someone had opened a fixed mortgage on a home for $500,000, and the housing value plummeted 50% (or more), the person may still have a fixed interest rate protecting the person from higher rates, but that doesn't protect the property value. In addition to that, if the person needed to move for a job, that person would face a difficult choice: move and sell at a loss, or move and rent and face some complications.
Renting is generally a good idea for people who (1) have not determined if they'll be in an area for more than 5-10 years, (2) want the flexibility to move if their living costs rises (which may be an issue if they lose wages), (3) don't want to pay property taxes (varies by state), homeowner's insurance, or maintenance costs, (4) enjoy regular negotiation (something which renters can do before re-signing a lease or looking for a new place to live). Again, other conditions can apply to people who favor renting, such as someone might enjoy living in one room out of a house rather than a full apartment or a person who likes a "change of scenes" and moves from one apartment to another for a fresh perspective, but these are smaller exceptions.
But with renting, you have nothing to re-sell and no financial asset so far as a property is concerned (thus why some real estate agents refer to it as "throwing away money" which isn't necessarily true, but one should be aware that the money they invest in renting doesn't go into an asset that can be re-sold).