forward rate under no arbitrage condition should be the rate implied by the rate with maturity T1 and T2.

if it is not, you your case,

1) forward rate is lower, which is f, the rate implied by 2-3 is r, f<r, you can long f and short r to get a risk-free profit.

2) f>r, the same reason

You forget the cash flow analysis last time I told you. For example, in case 1)

At time 0, you long a FRA contract and you also you borrow at a 3yr rate but lend at a 2yr rate.

2 years later, you get you 2yr lending back and invest in the forward rate agreement

another 1 year later, you get back the loan from FRA and pay back the 3 yr borrowing, you should get an additional cash amount.

this is a very neutral principle, I think, if you want to deposit 100$ for two years, at time 0 it is the same as you deposit 1 year and then deposit another year. (of course you have to lock the interest rate with FRA for the second year)
Roughly,

2 yr loan= 1yr loan + FRA(1-2 yr),

FRA(1-2 yr)=2yr loan- 1yr loan

best,