When I decided to get back on satellites again I had a 2m FM radio capable of 10 watts and a 2m squalo (horizontal omni) on the roof for transmit and purchased a Baofeng UV-3R for receive. I built a 'CJU antenna' with several feet of 12 gauge solid copper wire, a short piece of scrap pex pipe and a leftover end of a cable that had a BNC and RG-178 coax on it. (Coax used was only a few inches long) Antenna plans: http://personales.ya.com/ea4cax/paginaea4cyq/cju/cjuingles.pdf
I sit in my ham shack and work SO-50 with this setup... In fact, the CJU antenna mounted directly on my HT receives better at most times than the 7 element yagi I have mounted outside on a rotor; because the outside antenna is at fixed elevation and polarity. The important part of receiving SO-50 is the ability to aim and twist the antenna AND a very short piece of coax between the antenna and the receiver. (Or the antenna and a preamp.) Hearing SO-50 on an omnidirectional antenna... pretty much impossible. It has an ERP of 1/4 watt, hundreds of miles away and the satellite tumbles constantly changing the polarity of the signal.
The only other UHF downlink bird is FO-29... The other two current voice satellites are AO-7 and VO-52. AO-7 alternates between 2m and 10m downlink changing once a day. You can try listening to the 10m downlink with your HF rig, but there is not much activity. All the satellites planned to be launched in the near future are will all be 2m downlinks. And they will all be SSB/CW birds. You speak of using a converter... a lot would depend on that. I had trouble finding any useful converters for satellite when I started looking. I ended up with a single band all mode radio but quickly upgraded to an FT-847 which is designed for satellite. BUT that aside... a directional antenna held in the hand and aimed yourself will be FAR SUPERIOR to any omnidirectional antenna mounted on the roof. One, the gain. Two, less coax loss; a VERY big deal. Three, ability to compensate for tumble.
Guys are working satellites all the time with an HT and an Arrow antenna. Guys are working satellites with Yaesu FT-817s and an Arrow all the time. Two radios allowing you to receive your own signal while transmitting is the norm, a single radio is workable but has problems and is more likely for you to cause QRM. Guys do work satellites with eggbeaters and preamps, but they are also known as the guys who can't hear squat. They spend a lot of time saying, 'Who is that calling? Can you repeat your callsign?' A vertical or other omni might be acceptable for transmit even at 5-10 watts, but NOT receive.
Bottom line... if you insist on putting up an omnidirectional antenna, at the very least plan on a GOOD preamp and low loss coax, i.e. LMR-400. And then do not have high expectations. An Arrow would be far better... there are many pictures out there of antennas being hand held or mounted on tripods and they work much better than something on the roof. You don't NEED something on the roof... this is not terrestrial VHF/UHF, you are aiming into the sky. The only time height above ground is an advantage is if you want to try working passes that are close to the horizon and need to get above neighboring obstacles. With sats, far more important to be able to get away from obstructions like trees than to have height. In my case, my house is one story and a wooden roof; I sit in my hamshack chair and hear just fine. (I've seen a lot of installations with rotatable directional antennas mounted inside attics, by the way.) From your basement... would depend on the construction of the house. But with an HT, maybe you can go out on a patio or even try from upstairs.
If you don't want to buy, try building... here are a couple of designs:http://www.wa5vjb.com/references/Cheap%20Antennas-LEOs.pdfhttp://www.amsat.org/amsat-new/information/faqs/crow/
That second one is designed for even non hams and is made from cardboard and copper wire... THAT will be better for receiving UHF downlink birds than an omni on your roof... even with the latter using a preamp and low loss coax. - To quote Kevin Bacon in a Few Good Men, "These are the facts and they are not in dispute."
And please understand, no one is trying to be antagonistic, we have simply watched a lot of people try the 'omni on the roof' and waste a lot of time and effort... AND many go on to cause interference because they keep calling over the satellites causing QRM and don't know it because they can't hear the downlink.
When you get ready to try tracking and receiving the satellites, post a note over in the satellite forum and someone can recommend a program or a free website to help you do that.
73 es GL!