Government debt and Fiscal Theory of Price Level
Question:


  I understand that he is explaining 'non-Ricardian' policies. However,
  if the government increases its deficit by issuing bonds, would that
  not decrease AD in the period the bonds are sold? Say that the
  government increases its deficit in period t by selling bonds.
  Private agents buy these bonds and thus give up consumption goods. So
  in period t, should AD not decrease?


The act of issuing bonds does not directly affect AD, it only results in a rebalancing of portfolios: the household sector gives up money balances in exchange for government bonds. (If your model implies that spending depends upon money balances, there might be an indirect effect on the economy. In many models, money balances are zero, and so there is no effect. See below.) In any event, the end-of-period money holdings of the household sector depend upon the reaction function of the central bank within the model, and so that reaction function will effectively determine the final household portfolio allocation between money/bonds. (The reaction function is usually specified as a rule for the interest rate, which determines money holdings within the household's money demand function.)

The lack of relationship between money balances in macro models and transaction volume is not entirely obvious. The reason is that money balances are meant to represent end-of-period holdings, and transactions clear simultaneously. This means that a household with \$100 income can spemd \$100, and leave a \$0 cash balance at the end of the period. Since it is better to hold bonds/bills if interest rates are positive, the end-of-period money holdings woukd normally end up at zero. It can do this in every period, and so it is continuously transacting, yet measured money balances never appear. This is obviously unrealistic, and so other classes of models introduce reasons for end-of-period money holding. Two typical cases are a cash-in-advance requirement, or money in the utility function. The cash-in-advance requirement is that money holdings have to be at least some percentage of consumption spending (which simulates the reality that most wages are paid at a lower frequency than spending). Money in the utility function creates a demand for money within the optimisation problems, which usually depends upon the rate of interest. The end result is that the model assumptions imply a relationship between money holdings and the economy; change the assumptions, and the relationship changes.

Leeper is discussing a tax cut, which can trivially give rise to unchanged consumption (multiplier of 0). The government consumes the same amount as before, and thus runs a higher deficit (which is matched by greater bond issuance). If the household sector consumes the same amount, its savings rises by the amount of the tax cut, which it invests in bonds.

If the "multiplier" on the tax cut is positive, it implies that the household consumes more as a result of the tax cut, and so we have greater output. The exact effect depends upon the model you are using.

The question asks: 


  However, suppose that interest rates rose. If the interest rate is
  higher than the growth rate of the economy, the government will have
  to monetise its debt in order to pay the interest component, this will
  obviously be inflationary.


The quote from Leeper is a discussion of "Regime F." Within the analysis of the paper, monetising of debt has nothing to do with determination of the price level. The paper, the price level in regime F is given in equation (14), which is described: 


  At time t, the numerator of this expression is predetermined,
  representing the nominal value of household wealth carried into period
  t. The denominator is the expected present value of primary fiscal
  surpluses from date t on, which is exogenous.


That is, the expectation is that future primary surpluses will be used to stabilise the debt level.