This paper considers a downlink ultra-dense heterogeneous cloud radio access network (H-CRAN) which guarantees seamless coverage and can provide high date rates. In order to reduce channel state information (CSI) feedback overhead, incomplete inter-cluster CSI is considered, i.e., each remote radio head (RRH) or macro base station (MBS) only measures the CSI from user equipments (UEs) in its serving cluster. To reduce pilot consumption, pilot reuse among UEs is assumed, resulting in imperfect intra-cluster CSI. A two-stage optimization problem is then formulated. In the first stage, a pilot scheduling algorithm is proposed to minimize the sum mean square error (MSE) of all channel estimates. Specifically, the minimum number of required pilots along with a feasible pilot allocation solution are first determined by applying the Dsatur algorithm, and adjustments based on the defined level of pilot contamination are then carried out for further improvement. Based on the pilot allocation result obtained in the first stage, the second stage aims to maximize the sum spectral efficiency (SE) of the network by optimizing the beam-vectors. Due to incomplete inter-cluster CSI and imperfect intra-cluster CSI, an explicit expression of each UE's achievable rate is unavailable. Hence, a lower bound on the achievable rate is derived based on Jensen's inequality, and an alternative robust transmission design (RTD) algorithm along with its distributed realization are then proposed to maximize the derived tight lower bound. Simulation results show that compared with existing algorithms, the system performance can be greatly improved by the proposed algorithms in terms of both sum MSE and sum SE.