$100k won't buy you much more than a one-bedroom condo unit in Galleria ... and with monthly maintenance fees, it's hardly a cost-effective investment, especially if you plan to live in it a few years. Properties in West Houston are somewhat more affordable ... but still, you'd be likely looking at a townhome/condo in that price range, with the accompanying monthly fees. $200 and up is normal.
I've found the real values in the Copperfield and Champions areas. You can purchase a decent single-family home in the low $100's (figure around $120,000). They'll probably need a bit of work, but nothing major. At today's rates, you'd be looking at a mortgage payment of around $450 with 20% down. You can pay your taxes and insurance separately (somewhere around $3500), and there would be no PMI. If you wish to escrow them into your payment, add about $300 ... so you're in the $750 range. Depending on what you purchase, rents can range from $1200 to $1500 ... but understand that you'd be paying more for insurance (non-owner-occupied rates are higher) and you'd lose the homestead exemption, which would add about 20% to your tax bill ... probably in the neighborhood of $500. Overall, you'd still come out ahead, especially with the tax benefits.
Purchasing your new home while still owning the original property will likely require a written lease, in addition to a bit more documentation. You should definitely check with a lender to ascertain what you'll need.
I hope this helps. Please get back in touch, so we can further explore the possibilities. Talk soon ...