Learn English – Is “would + present tense” bad grammar

One way of tackling this problem is to comply with the rules of thumb that are presented in many grammars, stating that we have three (or four) conditional constructions: the (zero,) first, second, and third conditionals. Cambridge Grammar of English says:

The most commonly described conditional clauses are often known as the first, second and third conditionals. They have the following structures:
...

The second conditional
The structure of the second conditional is: if + simple past tense + modal verb with future-in-the-past reference (e.g. would/could/might).
In the second conditional, a speaker or writer responds to a possible or hypothetical situation by indicating a possible outcome. The speaker or writer states that the condition must be fulfilled for the present or future to be different:

• If I knew what you wanted, maybe I could help you.
• I would do a computer course if I had the time.

...

Your examples fall under the second conditional, and therefore they should be:

-Would it be alright if I went with you?
-Sure, it would be OK if you came with me.

This is one way to settle this, with an easy rule of thumb.
However, linguists don't seem to agree with that simplified categorization. They (or some of them) have rather an eccentric attitude toward this matter, which I'm not going to bring up, as I'm not a linguist and am not qualified to speak for them. But one thing we can do: we can carry out searches in corpora for conditional patterns and see if we can gain an understanding of the matter for ourselves.

A search in the Corpus of Contemporary American English for "would be ... if " (with a gap of zero to five words) yields 8665 results. (I searched for"would be" to exclude results like "would have", etc.) I examined the first and last pages (containing about 160 hits) and found about thirteen hits with the present tense in the if clause. Some of them are:

• We will die one day. I think the pride would be if you die with paving the way for others, with leaving something behind
• how dangerous it would be if executives try to shape newscasts to suit their business interests or ideological preferences
• What we're doing right now would be an absolute necessity if we are coming back in session.

Most of these hits were in the spoken category, then in the news, then magazines, and two or three in the academic category.

Some of the results were irrelevant, like this one:

• Well, the worst fear would be if that material melts down or catches fire

The same search in the British National Corpus yields just one or two present tenses in those 130 hits I examined, and in my opinion they're almost invalid:

• Some have seen this as illogical, and so it would be if the only reason for the rule that the release of one joint contractor releases the other is that there is only one obligation.
  _{('Is' is too far from 'if ... would')}
• Yes. Oh yeah we we if if it's if you're on a thirty percent and mo-- which most of them would be if you're on a thirty percent commission assignment then the other five percent just goes in the pot.
  _{(Too singular and incoherent)}

Although this hasty attempt is too shaky to qualify as a basis for extracting accurate, generic grammar rules, I think we can say these:
1- In American English the construction "would + present tense" is not very common in conditional sentences, but is used occasionally (on about 10% of the occasions). By a simple calculation, there should be over 500 instances(†) of this construction in COCA, which is a considerable number that should not be disregarded as wrong or invalid. Maybe it has special uses, but this is not the best place to investigate that. It's seldom used in the formal written style, for example.
2- In British English, "would + present tense" is almost never used in conditional sentences.
3- Overall, it may be safer to just stick with the so-called second conditional construction in exams or other sensitive situations.

(†) Edit: actually a lot more, because of the heavily restricted way of my searching

All that said, let me touch on the fact that the question in the title of your post is broader than the one in the body. I've heard some people use past tenses with would in non-conditional sentences, like "I wish that I would do you a favor that you actually needed". That's a whole other story, I think.