Hi,
As to the first problem - just check if you set default route on HUAWEI switches
ip route-static 0.0.0.0 0.0.0.0 <ip_of_mgmt_vlaninterf_on_CISCO_SW_L3>

The second issue...
Enterprise HUAWEI switches support 3 main modes of STP - stp, rstp, mstp (default)
If there is a problem witch compatibility related to other vendor sw, HUAWEI switch resorts to STP mode on the interface connected to the segment leading to the other sw.
PVST is proprietary CISCO protocol as far as I know, so you need to decide wether to change to MSTP mode on CISCO (and configure VLAN mappings, region settings, instances, priorities etc. on both HUAWEI and CISCO so MSTP on both switches groups can colaborate). That's the most preferable way, I think.
Not sure about pure RSTP mode on CISCO... You may try proprietary RPVST on CISCO and RSTP mode on HUAWEI but I haven't tested this mix so far. MSTP is supported by both HUAWEI and CISCO so it seems to be the best solution.
Check priorities on all switches and make sure, that CISCO is the root sw (if your topology sais it shoul be the root - but as it is L3 routing sw it seems it should)
You may also want to set :
stp pathcost-standard legacy
and on interface (trunk) :
stp no-agreement-check
(check documetation for ex. HedEx for "Setting Parameters for Interoperation Between Huawei and Non-Huawei Devices")

The other way is to set an ordinary stp mode on HUAWEI (stp mode stp), but that would be problematic in some topologies (non optimal paths, etc.) and is not recommended unless there is no other way.

Regards, Piotr