It's not a strangely specific list if you look at the GOP platform in each Presidential election year from 1976 through 2016. Starting in 1980, at the conclusion of the national "ideological sort", the common thread of GOP policy goals has been:

1. Reduce taxes and simplify the tax code

2. Appoint federalist pro-life judges

3. Privatize federal programs that can be privatized

4. Devolve funding and authority to the states through block grants

5. Oppose gun control

6. Require a balanced budget, and cut spending to do it

7. Subsidize school choice

8. Reform and cut welfare

9. Reduce the influence of unions

10. Increase defense spending

I didn't make those up; I literally went to the UCSB Presidency Project for each of those 11 cycles and made a text file capturing the policy objectives of the GOP (several months ago, not just as a freaky obsessive response to your comment).

There are other policy items that come and go; for instance, in the 80s and early 90s, the GOP was the party of term limits. Starting in the 90s, they became the party of tort, liability, and malpractice reform. For a brief, sparkling moment, they were the party of guaranteed-issue private health insurance. But those 10 items are basically unchanged, and none of them are Democratic goals.

To your specific arguments:

1. Obama opposed privatization and expanded federal entitlements dramatically.

2. Obama increased several taxes, and increased the tax burden on upper-income families and in support of the ACA; those increases would have been starker had he not been held hostage by a united GOP legislature that, for instance, required him to extend some of the Bush tax cats in exchange for funding the government. Not to mention: his biggest cuts were part of the ARRA, which prevented a potential second Great Depression.

3. Obama did not appoint pro-life or federalist judges of any sort.

4. Obama presided over the implementation of Common Core, which dramatically increased the federal influence over K-12 education and made him the bête noire of conservative educational wonks.

So, no, President Obama was not a "good Republican President". Respectfully, I think you might basically have to not know what the Republican party has stood for for the last ~40 years to believe that.

Fyi, I am not the commenter above who compared Obama to a "Republican president", and I tried to stick directly to your questions.

You'll note that the answers were mostly "no", except for whether he passed a major tax cut.