Without being able to see your site, it is hard to comment.
Many do not know that title length, either too short or too long, will effect if Google will use the title of not. As well, if the title does not seem to match the search intent, but another content element will such as the h1 tag, then Google may show that. There are several reasons why Google will not show a title the way the author intends. However, if you craft the title tag correctly, it will show the title correct most if not all the time.
Here are three links:
I have written more, but this should give you a good clue.
It is important to deploy a page with a good title right from the jump. Of course this is not always possible, but something you should strive for as much as you can.
You cannot hurry Google up. Sorry. Do not try. It makes the big G mad. You can encourage Google though by having fresh content. As a site ages and it remains fresh, Google will spider and index it quickly. I used to have a site that I could publish a page and have search traffic inside of 20 minutes. Freshness is the best way over time to have your site refreshed quickly.
Once a page is spidered, it is generally indexed right away. Google does not keep old copies of pages. It is simple semantics. It takes too much disk space.
It is not necessary that you increase the time for Google to spider your site. Google can really spider a site fast even with one request every 2 seconds. Google has spidered 78,000 pages of my site in a single day before. Also, keep in mind that Google goes in cycles sometimes. It may hit your site heavy and then barely at all for a while. Do not worry about this.
Any page that is fresh and has a history will be checked more often. Google uses a TTL (time to live - in seconds) style metric for individual pages. This means that any page that changes, Google will reduce the TTL and recheck it sooner including several times a day. Also keep in mind that there is an overall TTL for the site. So when a page is created, Google knows that the site is known for fresh content and will check it more often. The fresher your site is, the better response you will get from Google. But also keep in mind that just a few new pages a week can be enough. The website I told you about only had new pages added 3-4 per week.
I hope this helps. Let me know if you have any more questions and I will edit this answer.