Taking each part of your question separately Richmondlad ....
The third bullet point means your Clerk has authority to agree to expenditure but only if he/she consults with the Chairman of the Council or the relevant committee if your committees have delegated power to spend, for items up to the value of £1,000.00 The way the regulation is written implies it is blanket authority but most councils use this for emergency purchases or routine items such as stationery, etc. Such expenditure is not required to go through council as such as the financial regulations themselves in effect are the approval.
When payments are passed to the clerk to check that they are lawful, that is not giving the clerk authority to reject a decision made by council to purchase anything. My interpretation of that regulation is that when an invoice is received by the Clerk, he/she checks to make sure it is for something that the council (or, a relevant committee) has agreed to purchase, it is for the correct amount, is correctly addressed, etc. etc.
Councils without General Power of Competence do have to work to slightly different rules with regard to "lawful powers" and the clerk should be checking decisions to ensure there is a legislative power enabling the council to undertake that work etc. but this should be done before a decision is taken, not afterwards when payment is demanded for the work.
Just for clarity, legal responsibility for public footpaths remains with the landowner but of course many parishes have footpaths officers whose role it is to work with landowners and who sometimes undertake repairs and maintenance in the interests of the public. No one councillor can approve expenditure (i.e. the placing of an order) and then again, neither can two councillors place orders/purchases unless expressly authorised by the council to do so. However, the clerk, as an employee, can be authorised to place orders or agree expenditure (your bullet point above) so in your instance above, the clerk could have given you authority to spend that money whereas the chair was not empowered to do so.