Bonds are fixed-income assets. That's because once you buy a bond, you are guaranteed a fixed amount of investment income, for as long as you own it. The fixed amount of income is based on the coupon for the bond. Almost all bonds are issued for a face amount, say 1000 pounds, with a yield expressed as a percentage of the face value (also known as the coupon), and specified life of the security, referred to as the maturity.
So, a bond might have a par value of 1000, with a yearly coupon of 5% and a maturity of 10 years. An investor can buy the bond when it is first issued or buy it from another investor in the secondary market, at any time before maturity. IF the bond is purchased at issuance, the investor pays the bond issuer 1000. For the next 10 years, the investor receives a coupon payment of
1000*0.05 = 50 every year. After 10 years, the issuer returns 1000 to the bond holder. The investor has earned
50*10 = 500 over 10 years, for his investment of 1000. See here to understand more about how a bond's coupon rate differs from its market yield.
Bonds have various risks of default, as mentioned in the question. Bonds that are backed by the full faith of a government are close to risk-free, and provide a rate of return that is very close to the central bank's overnight funding rate. Corporate bonds are backed by the assets and management ability and product or service quality of the non-governmental organization, i.e. a business or company, who issues the bonds. Corporate bonds must pay a higher rate of interest than a government bond, otherwise investors would buy government bonds instead. The higher risk the company is, the greater the coupon (interest) it will need to pay in order to borrow money, i.e. in order to find investors who are willing to buy its bonds.
There are other entities that issue bonds, such as cities (municipal bonds) and electric companies (utility bonds). These bonds are less risky than corporate bonds, but more risky than national government bonds. That is because cities and utilities have a small but real chance of bankruptcy in the event of a natural disaster or terrible management.
Changes in interest rates affect the price of a bond. In the secondary market, the price of a bond varies depending on the yield curve. For a given credit quality, interest rates on shorter maturity bonds are usually less than on longer term bonds. If interest rates increase, the price of a bond must decrease. That's what is meant by a bond being a fixed-income asset. The coupon and face value of the bond are static, and remain the same from first issuance until maturity.
If a corporate bond's coupon is 5%, and the government raises the rate it offers on its bonds from 4% to 5%, then a corporate bond with a 5% coupon is not as good an investment anymore, as the investor isn't getting adequately compensated for the greater risk. (Central banks sometimes raise interest rates due to inflation.) As a result, the value of the bond is less, so it then must trade at a discount in order to remain competitive as an investment. In other words, the price of the bond at that point in time might decrease from $1000 to $950. The bond will still return 1000 at maturity, but in the interim, its price can be less, or greater than 1000.
If government bond rates decrease from 4% to 3%, then the corporate bond will continue to pay out at 5% but the price of the bond will increase.
Changes in credit quality can also cause the price of a bond to increase or decrease. If the bond was considered investment-grade at issuance, but fundamentals pertaining to the issuing company's status change and get worse, then the bond price will decrease even though the coupon rate stays the same.
I have omitted details about bond math, specifically, about bond yield. Note too that terms like "purely theoretically" or "in theory" aren't meaningful in determining the impact of an increase in government interest rates on a corporate bond. The only hard and fast rule is that bond prices go down when interest rates increase. However, other considerations such as improvement or decline in the bond issuer's credit quality or the overall shape of the yield curve (referred to as liquidity preference, i.e. usually people require a higher return if their investment is tied up for 30 years than for 1 year) can mitigate or amplify the impact of changes in government interest rates.