I’m a member of a fairly big United Methodist Church in suburban Chicago, and was on the church board for several years, so I have a limited amount of inside knowledge.
Yes, the budget is pretty much entirely funded by donations made by the congregation. If a church is fortunate, they may have had a few wealthy and/or generous members, who have willed significant amounts of money to the church.
A lot of parishes which have older buildings may own their buildings outright, which removes things like mortgages from their operating budgets. On the other hand, if the building winds up needing signficant renovations or repairs (as ours did), it can be a serious financial crisis for the parish. We had a multi-year fundraising program, with a lot of individual tactics (not quite “bake sales,” but close) to build up the nest egg which was needed for the renovations.
If a church is a member of a larger denomination, there might be some funds available to help struggling congregations, but generally, the cash flow goes the other way – individual parishes are expected to give a certain amount of money each year to the broader denomination, to assist with its operating budget. If, due to declining membership, and the commensurate decline in financial resources, a UMC parish is in dire financial straits, odds are high that it will end up having to close, or merge with another parish.
Our church is big enough, with a big enough congregation (and budgetary resources) that we have two full-time, salaried pastors (a senior pastor and a junior pastor), plus several other salaried employees (music director, church secretary, etc.). Conversely, I have some friends who are UMC pastors at small churches in rural Indiana – they are paid, but not very much, so they wind up having other weekday jobs to pay their bills, and they sometimes work as pastors for multiple parishes.
My understanding is that, in the case of the Roman Catholic Church, the diocese (the regional authority) usually is the owner of the church properties, rather than the local parishes. As church membership declines in many areas, many Catholic dioceses in the U.S. have been closing or consolidating parishes, often to the dismay of parishoners, though they often have very limited ability to fight against such closings, because the decision-making is being done at the diocese level.
The OP mentions Southern Baptist churches – denominations like the Southern Baptists are far less centralized than the Methodists or the Catholics, and the individual parishes are even more autonomous (and, thus, probably even more reliant on their own members to fund their budgets).