He is beautiful!
As far as neutering, that's up to you. It might help the humping it might not, some dogs stop as soon as they are neutered and plenty of neutered dogs hump too so I would not neuter if that was your only reason.
As far as age, again it's up to you. If you are completely confident he will not impregnate another dog (you will go the extra mile to ensure this) then I do think it is healthiest to wait until maturity (about 2 years old) to neuter. That way the body gets to make of use of all the right hormones while it is growing. This allows his body to fully mature as it is meant to and prevents some health problems. Fixing him before he gets more elderly prevents other illnesses. Marking in the house is not as common in intact dog as it would be in cats (in cats it's almost a guarantee) so I would not really expect it just because the dog is intact. If he starts treat him like a housebreaking puppy and start right away so a habit does not form.
Humping is normal but should be discouraged as I've heard dogs can hurt themselves (can't get "it" back in). I agree with redirecting him when or when he looks like he might hump. If he never gets to do it (you always stop him almost immediately) he will find it less enjoyable.
I would not personally get the lyme shot for my dog unless I hiked all the time in an area full of ticks and had a dog with a coat that made it difficult to find ticks. So long as you give him a really good check after going into a field or in the woods you should be pretty safe. I would not risk the vaccine (but I m not a huge fan of vaccines in the first place).
