Update Qiskit 0.35 introduced a new gate XXPlusYYGate.
$$\newcommand{\th}{\frac{\theta}{2}}\\\begin{split}R_{XX+YY}(\theta, \beta)\ q_0, q_1 = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & \cos(\th) & i\sin(\th)e^{i\beta} & 0 \\ 0 & i\sin(\th)e^{-i\beta} & \cos(\th) & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}\end{split}$$ So, you can now add parameterized $\text{XY}$ to your circuit as follows:
theta = Parameter('θ') circ.append(XXPlusYYGate(theta, 0), [0, 1])
Original Answer
For the first part of your question, we have $$XY(\theta) = \exp(-i {\frac{\theta}{2}} (X{\otimes}X + Y{\otimes}Y))$$ And since $X{\otimes}X$ and $Y{\otimes}Y$ commute, we can write it as $$XY(\theta) = \exp(-i {\frac{\theta}{2}} X{\otimes}X) \exp(-i {\frac{\theta}{2}} Y{\otimes}Y)$$ Qiskit already has these two gates: $$R_{XX}(\theta) = \exp(-i {\frac{\theta}{2}} X{\otimes}X)$$ And, $$R_{YY}(\theta) = \exp(-i {\frac{\theta}{2}} Y{\otimes}Y)$$
Hence, the implementation of $XY(\theta)$ as a parameterized gate in Qiskit will be as simple as
from qiskit import QuantumCircuit from qiskit.circuit import Parameter theta = Parameter('θ') circuit = QuantumCircuit(2) circuit.rxx(theta, 0, 1) circuit.ryy(theta, 0, 1) param_iswap = circuit.to_gate()
Another Solution
If you want to use more basic gates than rxx and ryy, you can use Qiskit's Operator Flow:
H = 0.5 * ((X^X) + (Y^Y)) theta = Parameter('θ') evolution_op = (theta * H).exp_i() # exp(-iθH) trotterized_op = PauliTrotterEvolution(trotter_mode = Suzuki(order = 1, reps = 1)).convert(evolution_op) circuit = trotterized_op.to_circuit() circuit.draw('mpl')
The composition: 
And as before
param_iswap = circuit.to_gate()
For the second part of your question, I think the best answer you can have is the one mentioned in the comments by @epelaaez, as it is recent and from a member of Qiskit's development team.
