If a church member(no one on staff or on the board) has serious car trouble, needs a new engine but is struggling financially and can't afford one, can the church as a means of charity, give this person the money to replace the engine?

Charity to individuals is a dangerous thing. It can be done - but (a) it's a taxable payment to the individual (the church needs to issue a 1099-MISC if the award is greater than $600), and (b) as a matter of good governance the church board better have a very clear policy on the criteria for making such awards. How do they determine who is needful enough, and in what amount? Do they give money for car repairs to anyone who asks? Who makes the decision? Also, if this expenditure was not included in the approved annual budget the full board ought to take a vote to approve it.