Best way to deal with UK immigration - US citizen

I'll start by saying it's generally a bad idea to try to shorten the landing interview by bending the truth or making something up. Time and again people come up with ways to get around an awkward situation and that'll only sink your friend faster. As noted in the comments, your friend will be questioned, the only variable is how much. As a matter of opinion (and of a recent question on this site) I would also advise against assuming that he'll be granted entry painlessly because of his citizenship.

I expect he'll be grilled about these topics.

Premise

It's the border official's job to be suspicious, your friend has no premise for being there. A recently unemployed person suddenly decides to stay in the UK for a few months ? Seems unlikely.

Home ties

Given that he's unemployed, he'll need convincing evidence of solid ties. Something like property ownership, upcoming medical appointment or family. They want to know he intends to leave, and as you've noticed, a flight to mainland Europe is easy enough to dispose of so as not to count very much. This answer could be helpful reading about home ties.

Finances

He has savings, but he'll need to prove he can cover his entire stay there and his return ticket, seeing as he doesn't have one. Note that according to this answer having a credit card might not be enough. Although the answer in the link talks about visas (which he doesn't need) I think they're a good guideline to proving his spending pattern won't ruin him while in the UK.
They'll also ask where the money came from, this question talks about that.

Relationship to you

Even though you are not in a relationship, the burden of proof is on him. The length of the stay will look suspicious and he'll need to convince the border official he's not trying to move permanently. They will ask details about how you met, how long you know each other etc. The fact that he has no previous travel history will not help him because these are sometimes used to establish patterns of travel.

Addendum

Even though US nationals aren't required to get entry clearance before arriving to the UK, doing so might simplify his entry once at the border. In the case of refusal, at least he'll avoid the hassle (and possible removal) you don't want him to experience. Read here for how to apply.

Your current plan, a long term visit without a definite end date, is one that is going to cause questions. A multi-month absence means his life in the US is put on hold, which implies weak ties. Regardless of photos, he is prepared to leave his dog for a substantial slice of its life.

You are considering pretending a different plan that would look more normal. Have you considered making the pretense reality, so that he neither lies nor presents a suspicious appearance?

Unless you have already spent a lot of time together, a multi-month visit is not that good an idea anyway.

In this strategy, he would plan to spend two or three weeks in the UK, as a tourist. He would buy a return ticket, reducing his air fare. He could either stay with you or stay in a hotel, but in any case would have evidence of sufficient funds to support himself in the UK for the known duration of his visit.

Meanwhile, he would continue with his life in the US. Look into jobs or education, but schedule interviews and/or enrollment for after his UK vacation. When he applies for entry, he would know what he plans to do on returning to the US.

As it currently stands on the balance of probabilities (since this is the civil test the IO will apply) there is a chance that he'd be refused entry since he is unemployed, no return date, no immigration history, visiting a female friend in the UK...if he is upfront then you never know he might be landed. IOs appreciate frank and honest people. He might even be TA-ed (Temporary Admission) for a few days even if he is refused Leave to Enter (LTE).

On the other hand if you add deception into the mix then he will be refused LTE and as pointed out already NEVER ever deceive an IO they will get to the bottom of his intentions in any event. He'd make matters much worse for himself. Just tell him to be honest about his intentions if they ask him at the border.