Spreading lime and fertilizer at the same time is not recommended. The reason is there will be a reaction between the two and plant nutrients will be lost to the air. As a rule you should wait 1 month after liming to apply fertilizer. But in your situation I would have to make an exception, due to the time constraints and distance.

It is not recommended to apply more than 3 ton of lime per acre, in one year. It can/will "shock" the soil.

There are many kinds of store bought PH testers available, the probe meter type are junk in my opinion. The ones that are in a kit, were you can do individual test for PH and N-P-K are more accurate but fairly expensive.

Sending soil samples to a lab is hard to beat especially for a new piece of ground or food plot. They will do detailed exam of your soil, let you know what is lacking and what and how much is needed to correct any problems, saving you money in the long run. IMHO.