Should I use singular or plural here?

A plural verb-form here doesn't work well. There is a fixed phrase

• This appears to be why ...

for instance

• This appears to be why Ian Rapoport was suspended
• This appears to be why Senate Majority Leader Mitch McConnell reconvened the Senate on May 4
• This appears to be why they improve insulin sensitivity and reduce body fat ...
• Indeed, this appears to be why particle theorists were so off base in their prediction

(examples from 144 000 Google hits), while

• these appear to be why

has one relevant hit. This apparently reasonable variant (there could easily be several reasons involved) seems unwelcomed, perhaps because it sounds incongruous. An easy override explanation by those using notional agreement is to see the multiple reasons as adding up to one compelling situation (with anaphor 'This'). But to my ears, OP's second variant doesn't sound too well-formed either.

As

• Jack and Jill appear to be getting on well, considering and
• This, and the fact that my wife works for an animal charity, appear to be the major reasons why I have six dogs

are idiomatic and obviously grammatical, it looks like acceptability or otherwise here isn't a matter of grammar.

Adding padding to the original as in the last example above, or it-clefting:

• I like to adopt abandoned things. It appears that this, and the fact that my wife works for an animal charity, are the reasons why I have six dogs

restore idiomaticity.

The second approach is ok to me.

Also, I have another way which I think it could match your mean well.

It appears that both my hobby of adopting abandoned things and the fact that my wife works for an animal are the reasons why ....

I think it'd be better not to use the commas in the first place as follows:

I like to adopt abandoned things. This and the fact that my wife works for an animal charity appear to be why I have six dogs.

Here, This and the fact that my wife works for an animal charity are syntactically combined into a plural subject, thus requiring plural verb appear.

Not all writings are well thought out, however, so some writers might start with This and the fact that... without a comma after This, thinking that they're presenting a plural subject, only to add a comma before plural verb appear to somehow separate the that-clause from the following main clause:

I like to adopt abandoned things. This and the fact that my wife works for an animal charity, appear to be why I have six dogs.

Some might leave it as is, resulting in a bad writing, and some might try to fix the bad writing and, as an afterthought, add another comma after This to make it look better:

I like to adopt abandoned things. This, and the fact that my wife works for an animal charity, appear to be why I have six dogs.

Now they didn't intend to make and the fact that my wife works for an animal charity a supplement, but nevertheless ended up making it look like one by adding commas around it. The problem is, then, they don't realize they essentially made it look like a supplement, or they might even be thinking that it's not a supplement even though it has commas separating it from the rest of the sentence.

But the reader has no way of figuring out what the intended structure was, other than relying on the commas, necessarily leading to confusion.

In order to avoid this confusion, you can either not use the commas at all as suggested in the first rewrite above or rewrite the text altogether as follows:

I like to adopt abandoned things. Because of this and the fact that my wife works for an animal charity, it appears, I have six dogs.

Since there seems to be considerable disagreement over what is proper, what might be acceptable, and what justifications exist, I'm going to post a few examples of this construction as I find them. This is a work in progress.

Characterization of Freeze-Thaw Damage Mechanisms in Composites for Civil Infrastructure, Haramis et al. p401)

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Painting Out of the Ordinary, Solkin, p106

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Hearing to Examine the Occupational Safety and Health Administration's Reinvention Project

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Environmental Imapct Report for the California-Oregon Transmission Project (I typed this one because the imaged text was turned.)

This, and the fact that bald eagle use of the areas around the crossings is relatively low, suggest that the potential for collisions is relatively low.

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Nature As Teacher New Principles in the Working of Nature By Viktor Schauberger, Callum Coats · 1999

This, and the fact that he developed free energy machines through harnessing the magical processes of Nature, has made Viktor Schauberger truly a man of our times.

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Japan's Construction by Kaigai Kensetsu Kyōkai (Japan) - Page 31, 1963

This and the fact that the steady flow of people into urban areas has caused serious pollution of almost all rivers have made people in Japan realize keenly the importance of adequate sewerage.

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It would seem that both are used by professional writers.