Yes, you can use "разрешать" instead.
Yes
In this case "по вашему проекту" means "about your project" or "related with your poject". Examples:
"У меня есть несколько вопросов по вашему проекту" - "I have few questions about your project"
"Есть какая-нибудь информация по нашему проекту?" - "Is there any information about our project?"
If you use "какой-то" meaning "some" - you show, that you lack some information about the entity you are refrerring to. For example:
"Если у вас есть какие-то вопросы - пожалуйста, задавайте" - "If you have any questions, feel free to ask" (you don't know what questions would be asked)
"Какие-то люди вошли в комнату" - "Some people entered the room" (You don't know these people)
"Какие-то мысли крутились в голове" - "Some thoughts were on my mind" (You can't clearly explain what these thoughts were about)
But in this case you know exactly what you want to say, so you can't use "какой-то".
I won't say there's to much doubt in this question. Your phrase is correct, but "старых шин" ("old wheels") isn't what you wanted to use, I think. "Подержанных шин" ("used wheels") is much better.
There's contraction with ablative case "Будучи финансовым конcультаном".
"From the account" would be "со счёта". "За счет" is "by the means of". Is refers to the source of finance. "Образование финансируется за счет федерального бюджета" - "education if financed from federal budget"
Yes, "поддерживаю позицию" or "разделяю позицию" is correct.
In common, we don't use "основываться на" in case of financial resourses.