What others said.
None of this goes to Rome.
The diocesan bishop erects (and suppresses) parishes, alters their boundaries, etc, while taking account of needs, resources, viability, local wishes, etc as best he can. He also decides what priests will staff parishes.
As regards financing the construction of the church — in theory a parish is supposed to be financially self-sustaining, and so able to supply its own church. In reality new parishes are usually established with the expectation that they will, in time, become financially self-sustaining but they aren’t initially, and the diocese usually assists in funding the construction of a church for a new parish. But that may be partly by way of loans that the new parish is expected to pay off over time.
As regards art, design, etc — there are rules and guidelines in canon law about what should or should not be in a church, how the various parts of the building should relate to one another, etc. Ecclesiastical architects will generally be familiar with these. There’ll also be a budget for the project. After that, on all decisions the ultimate authority is with the bishop, but he’s someone with many calls on his time and on an issue like this he will usually satisfy himself with approving decisions made by others. The more the parish is paying for its own church, the more likely it is that the bishop will defer to the wishes of the parish on matters like this. The parish priest/parish council/a committee assembled especially for the purpose will usually be the primary point of contact with the architect, builders, contractors, etc.
Apart from parishes, there’s also the matter of monasteries, convents and religous houses of various kinds that often provide parish-like services to the communities in which they are situated. These are provided and operated by religious orders, The bishop has less of a role here. A religious order needs the permission of the local bishop to establish a house in his territory but, once that permission is given (and not withdrawn) the financing, operation etc of the house is their business, not the bishop’s. As a matter of practice they may co-operate closely with the bishop/the diocese, but that’s on the basis of mutual agreement, not because one of them is subordinate to the other. In general the religious order concerned finances, designs and builds any church or chapel that may be included in the house. The diocese may assist with financing or other material support (e.g. providing a site) if there is an agreement that the religious order will do things that, if they weren’t there, the diocese would need to do — operate a school, staff a parish, that kind of thing.