First, you are not forming a partnership, instead, you are forming a limited liability company. Stop using the word partnership because it has a specific legal meaning and the use of the word may lead to greater liability than you intend.
Second, what do you mean by the LLC being under your name? An LLC is an entity in which you own a membership interest. It is a separate legal entity. You, as a member may be either a managing or a non-managing member. An LLC also has an option of hiring managers that do not have a membership interest.
Not having any additional information, this is how I would answer your questions:
1 & 2. An LLC will have separate finances, however, without an established credit rating for the company, you will not find a landlord who will lease to an LLC with out personal guarantees from the members. Any decent lease will make each member liable for the entire amount of the lease. So if one of you three dies or goes bankrupt, the other two would be on hook for the whole amount--effectively, the lease will force you into a partnership for the limited purpose of renting a space. If your company persuades a landlord to lease with out personal guarantees, it does not matter where the money comes from, it would probably be treated as a loan or a contribution, but it would not make you any more liable.
3. You will not enter into a partnership agreement, instead, you will enter into an operating agreement. If drafted well, an operating agreement will provide for ways to deal with arising problems and govern each member's interests and obligations. It will also provide for an amendment mechanism to help overcome unforeseen developments.
While forming an LLC does not require an attorney by any means, it is highly advisable that you consult an attorney and a CPA to figure out how things work and to make sure you receive the full protection and benefits available to an LLC.