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I'm trying to understand if my rationale behind discount bonds is correct.

Let's say Company X issues a 2 year $1000 bond at 5%, market interest rate is 10%. Because the coupon rate < the market interest rate, Company X is going to receive strictly less than $1000 for the bond, let's say ($1000 - a).

Now is the rationale behind this because investors could keep their $1000 instead, make 10% off market interest, and so they wouldn't be willing to loan for 5% interest? Therefore, they pay some amount < $1000. Is this correct?

Moreover, how much does the bondholder actually receive in total at the end of 2 years? Does he receive the PV of future cash flows of the principal ($1000) and the interest ($1000 * 2.5% semiannually)?

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It's simple. If the market rate is 10%, all else being equal, the investor would expect to see a 10% YTM (Yield to Maturity) on the investment. It is either a return via coupons (interest) or principal at the end of the term.

If the coupons are $100, the issue price would be 100, (i.e. 100% or $1000 for this bond). A zero coupon bond would be $826, and the 10% compounds to reflect the $1000 payment at maturity.

Any coupon lower than market necessitates an initial price below 100.

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